Chapter 5
Multiplication and Division I: Meaning
5.1 Multiplication as Repeated Addition
Multiplication is not really a basic operation. As the problems in the following activity show, it is possible to solve
many “multiplication” problems by using a simpler operation.
Activity 5.1A
A. Solve the following problems using addition and appropriate units. Draw pictures if it is helpful to do so.
1.
Three children are playing a game. Each child gets four cards. How many cards are in use?
2.
A rectangular baby quilt is made of four strips each containing six squares. How many squares are in this quilt?
3.
Rachel has two pairs of shorts and three T-shirts. Assuming she is indifferent to color coordination, how many
outfits does she have?
4.
A water bottle has a capacity of 11/2 liters of water. How many liters of water can five of these bottles hold?
B. Answer the following.
1a. Each of the problems in part A involved repeated ___________________ .
b. Each of the problems in part A could have been solved more efficiently using what operation? _____________
c. Thus multiplication can be defined as __________________________________________________________
2.
Consider the following sets.
♥
♥
♥
♥
♥
♥
a. There are _____ sets with ______hearts in each set. The union of these sets includes six ____________.
b. In other words, 3 ∙ (2 __________) = 6 _______________
c. In this problem, 3 refers to the ___________________ of sets and 2 refers to the ___________ of a set.
3.
Reconsider problem #4 in part A. Five referred to the _______________ of bottles and three quarters of a liter
referred to the _______________ of a bottle.
4.
In these situations, it seems that one of the numbers in a multiplication refers to the _____________________
and the other refers to the _______________________________.
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In all of the above problems, answers can be found by using repeated addition. There are so many situations
involving repeated addition that this process is called multiplication. (Be warned, however, that repeated addition is
not the only meaning of multiplication. We will study another meaning in a later section of this chapter.)
Basic Definition of Multiplication as Repeated Addition
For m a whole number, the product m • B is the total number of objects in m disjoint sets, each
containing B elements. m is called the multiplier and B is called the multiplicand.
m•B=
B
+
B
+
B
+
...
+ B
m times
The two numbers m and B play two very different roles in this basic meaning of multiplication. The multiplier m is
the number of sets while the multiplicand B is the size of the set. The result of a multiplication is called a product.
In situations in which multiplication is defined as repeated addition, the multiplicand can be any type of number but
the multiplier must be a whole number.
Total = (Number of sets) • (Size of the set)
↓
↓
↓
Product = Multiplier
• Multiplicand
Example 1:
Melissa invited all of her running friends over for a morning run followed by brunch. She bought
three dozen eggs for the occasion. How many eggs did she buy?
Total number of eggs = 3 sets of 12 eggs = 12 eggs + 12 eggs + 12 eggs = 3 · 12 eggs = 36 eggs
“Of” and “Times”
Notice that “of” is the word we often use to describe the size of a set. For instance, we might say that a platoon
includes three squads of 10 soldiers. This phrasing indicates that the total number can be found by repeated addition,
a.k.a., multiplication. IThus the use of the word “of” can be a signal to multiply. Conversely, “times” can often be
translated as “of.” For example, “3 times 5” can be interpreted to mean “3 sets of five” or 3 fives.
Teaching Tip: Sometimes children are told that “of” means“times.” This is a misleading overgeneralization.
“Of” is one of the most common words in the English language and often does not mean “times.” For example, in
the following sentence, “Nine of the 12 students in the class passed the test,” it would be nonsensical to multiply 9
by 12. It actually makes more sense to say that “times” often means “of.”
Factors and Multiples
The multiplier and multiplicand are also called factors. A whole number product is called a multiple of each factor.
Example 2:
Consider 3 · 2 = 2 + 2 + 2 = 6.
a. 3 is the multiplier, 2 is the multiplicand, and 6 is the product.
b. 2 is the size of the set, and 3 is the number of sets.
c. 3 and 2 are factors of 6, while 6 is a multiple of 3 and 2.
Every whole number except 0 has a finite number of whole number factor, but all whole numbers have an infinite
number of whole number multiples.
Example 3:
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Set of factors of 6 = {1, 2, 3, 6}; set of multiples of 6 = {0, 6, 12, 18, . . .}
Teaching Tip: Students often confuse factors with multiples. For instance, a student might say that 3 is a multiple
of 6 or that 12 is a factor of 6. Since these are important vocabulary words, teachers need to spend time making
sure students learn which is which. Mnemonic devices such as “Factors are first” or “Multiples multiply
monotonously” may be helpful to some students.
As the next examples indicate, many different notations are used to indicate multiplication.
Example 4:
(a) Product of 2 and 3 = 2 times 3 = 2 threes = 2  3 = (2)(3) = 2(3) = 2 * 3 = 2 • 3
(b) Product of x and y = xy = x • y
Units in Repeated Addition
A sum has the same unit as its terms. For example, 3 feet + 3 feet is equal to 6 feet. Similarly, since the basic
meaning of a product is the repeated sum of multiplicands, the product has the same unit as the multiplicand.
Example 5:
Five yardsticks are placed end to end. How many feet is it from one end to another?
5 • 3 feet = 3 feet + 3 feet + 3 feet + 3 feet + 3 feet = 15 feet
Activity 5.1B
A. Fill in the blanks, representing the total as a repeated addition. Include units.
Multiplier Multiplicand
Ex: Three days a week Heidi walks 13/4 miles.
How many miles does she walk every week?
3
13/4 mi.
Total
13/4 mi. + 13/4 mi. + 13/4 mi. = 51/4 mi
1.
Sara has two classes of 20 students. How
many students does she have altogether?
_____
________
_______________________________
2.
Peter buys three ½-gallon bottles of milk.
How many gallons of milk has he bought?
_____
________
_______________________________
B. Answer the following questions.
1a. Find the area of the shaded shape on the centimeter grid to the right.
_________
b. What is the shape of the standard unit for measuring area? __________________
2a. Suppose each cube to the right measures 1 cm by 1 cm by 1 cm.
What is the total volume of this set of cubes?
_________
b. What is the shape of the standard unit for measuring volume? _______________
Four Major Situations Involving Repeated Addition
1. Distinct Repeated Sets
Example 6:
Consider the problem in which each of three children has four cards. How many cards are there
altogether?
We have three sets of four: 3 • 4 cards = 4 cards + 4 cards + 4 cards = 12 cards.
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The most obvious case of repeated sets occurs with a repeating set of physical objects. This physical set may be a
hand of cards, a platoon of soldiers, a case of soft drinks, and so on.
2. Arrays
Consider the situation in which Rachel has three T-shirts and two pairs of shorts. The following diagram illustrates
one way to determine that Rachel can put together a total of six different outfits.
A horizontal arrangement of objects is called a row and a vertical arrangement is called a column. The above
diagram, with 2 rows and 3 columns, is an example of a 2 by 3 array. An R by C array is a set of discrete objects
arranged into R rows and C columns. Because the rows of an array are the same size, the total number of elements in
an array can be found by repeatedly adding the rows. Since the row size is the same as the number of columns, we
have the following generalization.
The total number of elements in an R by C array is R • C.
This explains why an R by C array is also described as an “R  C array.”
Example 7:
This is a 2  5 array, with two rows and five columns.
Total number of elements = 2 • 5 = 5 + 5 = 10
۞ ۞ ۞ ۞ ۞
۞ ۞ ۞ ۞ ۞
3. Area and Volume
What is the total number of squares in a baby quilt made of four strips of six squares each?
This is another example of a problem that can be solved by repeated addition. The quilt
consists of four rows, each containing six squares. The total number of squares is equal
to the following: 4 sixes = 6 squares + 6 squares + 6 squares + 6 squares = 24 squares.
This quilt also illustrates why the area of a rectangle can be found by multiplying its length by its width.
Finding the number of squares in a rectangle is analogous to finding the number of elements in an array.
Rectangles as Arrays of Squares
Array with 8 elements
Rectangle with an area of 8 squares
Generally speaking, we measure the area of a two-dimensional shape using squares. The squares in a rectangle form
an array in which the number of rows corresponds to the length of the rectangle, while the number of columns
corresponds to the width. Thus the area of a rectangle is the product of its length and width.
B
Formulas for the areas of other special shapes are derived from this basic area formula.
H
Example 8:
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The area of a right triangle with legs of length B and H is ½BH because
its area is half the area of a rectangle with length B and width H.
One special area is not directly derived from the area of a rectangle. The area of a circle is equal to π r 2, where r is
the radius of the circle.
As the following example illustrates, the area of many figures can be found by partitioning the figure.
Example 9:
To find the area of the figure given below, partition it as indicated.
6 cm
Area Half-circle = 0.5 π (3.8 cm)2 ≈ 22.68 cm2
6 cm
3.8 cm
Area Rectangle = 6 cm · 7.6 cm ≈ 45.6 cm2
7.6 cm
16.8 cm
3.8
6.0
7.0
Area Triangles = 2 · (0.5 · 3.8 cm 7.0 cm) = 26.6 cm2
Area Total = 94.88 cm2
Volume
1″
The standard unit for measuring volume is a cube. A cube that measures one unit
by one unit by one unit has a volume of one cubic unit. As the following activity
illustrates, the volume of the three-dimensional analog of a rectangle can be found
by repeated addition of layers of cubes.
1″
1″
One Cubic Inch
Activity 5.1C
1.
A solid box has a length of 4 cm, a width of 2 cm, and a height of 3 cm.
________ a. What is the area or the bottom (or top) of this box?
________ b. How many cubic centimeters are in the first layer of this box?
________ c. How many layers does the box have?
________ d. Use the above facts to determine the volume of the box.
2.
What is the volume of a box that is 5'' high, 10'' long, and 3'' deep?
______________
3.
A cylindrical water tank is 20 feet high. It is known that when the water is one foot deep,
the volume of water in the tank is about 700 cubic feet. What is the capacity of the tank?
[Hint: Think about the volume of each layer.]
_____________
The formal name of a typical box is a right rectangular prism. It has rectangular faces
at right angles to each other. A right rectangular prism with length L, width W, and height H
can be partitioned into a series of identical one unit thick layers. The volume of one of these
layers has the same numerical value as L· W, the area of the “floor” or base of the prism.
Since the number of layers corresponds to the height of the solid, the volume of the right
rectangular prism is as follows.
1
1
1
1
W
L
Volume of a right rectangular solid = length • width • height
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Volumes of Solids with Congruent Bases
In general, a prism is any solid with two congruent and parallel polygonal bases connected by parallel lines. This
means that the other faces of a prism are parallelograms.
Various Prisms
A prism is a special type of cylinder. A cylinder is any solid with two congruent and parallel bases, not necessarily
polygonal, that are connected by parallel lines.
Various Cylinders
Like a prism, a cylinder consists of a series of congruent layers. Thus its volume is the repeated sum of the volume
of one layer. The volume of a single layer has the same numerical value as the area of the base of the cylinder; the
number of layers corresponds to the height of the cylinder. (The height of a cylinder is the distance between its
bases. If the base of a cylinder is horizontal, then its height is vertical.) This yields the following useful formula.
Volume of a Cylinder = Area of its Base • Height
Example 10:
If the base of a kidney-shaped pool has an area of 40 square feet, then filling it to a depth of one foot
will require 40 cubic feet of water. Every additional foot of depth will require another 40 ft 3. So
filling the pool to a depth of three feet will require 40 ft3 + 40 ft3 + 40 ft3 for a total of 120 ft3.
Example 11:
A waste basket is a cylinder that is 2′ 3″ high. Its base has parallel
sides and circular ends. The parallel sides are 10 inches apart and
one foot long. How many gallons of water will this waste basket
hold? There are 231 cubic inches in a gallon.
Find the area of the base. It consists of two half-circles and a rectangle.
The area of a circle is  r2, where r is the radius. In this situation, the
diameter is 10″ and thus the radius is 5″. To reduce round-off error,
do not round until the end of the problem.
12″
10″
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Area of rectangle = 10''  12'' = 120 square inches
Area of two half circles = 2  (½  r2), where radius is 5″
 3.14159…  52 square inches
 78.5 square inches
Total area of the base
 198.5 square inches
Volume of container




198.5… square inches  27 inches
5360.57… cubic inches
5360.57… in3  231 in3 per gallon
23.2 gallons
4. Cartesian Products
Recall that the number of possible combinations of Rachel’s shorts and T-shirts was found by pairing each T-shirt
with a pair of shorts. In general, the set consisting of all possible ways of pairing elements of a set A with elements
of another set B is called a Cartesian product. A Cartesian product can always be illustrated as an array. The
number of rows in this array corresponds to the number of elements in set A, designated as NA, and the number of
columns corresponds to the number of elements in set B, designated as N B. Thus we have the following.
If C is the Cartesian Product of A and B, then N C = NA • NB
Example 12:
The license plate of a very small state consists of a letter followed by a single-digit number. How
many distinct license plates of this description are possible?
The license plates form an array, partially indicated below.
0
1
2
3
4
5
6
7
8
9
A
A0
A1
A2
A3
A4
A5
A6
A7
A8
A9
B
B0
.
.
B1
.
.
B2
.
.
B3
.
.
B4
.
.
B5
.
.
B6
.
.
B7
.
.
B8
.
.
B9
.
.
Z
Z0
Z1
Z2
Z3
Z4
Z5
Z6
Z7
Z8
Z9
There are 26 rows with each row containing 10 plates. The total number of plates is 26 • 10 or 260.
A Cartesian product can also be described using a tree diagram, as shown below.
Example 13:
Let S represent a pair of Rachel’s shorts and T represent a T-shirt. The following tree diagram shows
the six outfits that result from using these clothes.
S1
S2
T1
T2
T3
T1
T2
T3
S1T1
S1T2
S1T3
S2T1
S2T2
S2T3
As the next activity demonstrates, the idea of a Cartesian product can be extended to more than two sets.
Activity 5.1D
1.
Find the volume of a prism that is one foot long with a right triangular base.
The three sides of the base measure 3'', 4'', and 5''.
2.
Suppose license plates consist of a letter followed by two digits.
_________________
a. List one license plate meeting this description.
_________________
b. How many license plates meeting this description start with A?
_________________
c. What is the total number of license plates?
_________________
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3. In Tennessee, license plates consist of three letters followed by three digits.
a. How many license plates are possible in Tennessee?
_________________
b. Suppose Tennessee deletes 38 three-letter words from use on license plates.
How many license plates are now possible in Tennessee?
4.
________________
Summarize the pattern that occurs when a fraction is multiplied by a whole number in the following.
a. 4 · 1/2 = 1/2 + 1/2 + 1/2 + 1/2 = 4/2
5.
b. 3 · 4/5 = 3 · 4 fifths = 12 fifths = 12/5
c.
2 · 7/3 = 7/3 + 7/3 = 14/3
Use the pattern you observed in the previous problem to find the answer to the following word problem.
A chocolate nougat weighs 2/3 ounce. How much do 5 of these nougats weigh?
The set of all possible Tennessee license plates is an example of a general Cartesian product. Just as a license
plate is created by choosing letters and digits, an element in a general Cartesian product is formed by choosing
elements one at a time from several sets.
N1
elements
N2
elements
N3
elements
N4
elements
Set 1
Set 2
Set 3
Set 4
. . .
Nk
elements
Set k
General Cartesian Product
Each element in this Cartesian product contains one element from Set 1, one element from Set 2, and so on. The
total number of such elements is found as follows.
Total number of elements in the Cartesian product = N1 • N2 • ...• Nk
Example 14:
How many different kinds of pizza can be made if there are five possible toppings from which to
choose?
For each topping, there are two choices, to use the topping or not to use it. Thus there are a total of
five sets, each containing 2 choices. So the total number of pizzas is equal to 2 • 2 • 2 • 2 • 2 or 32.
The next example illustrates a situation in which several sets need to be reconsidered as a single set in order to
determine the appropriate number of possibilities.
Example 15:
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Suppose Tennessee license plates consist of three letters followed by three digits, with 38 three-letter
words deleted from use. How many license plates are possible?
Total number of allowable “words” = 263 - 38 = 17,538.
For each word, there are 103 or 1,000 numbers.
This yields: 17,538 • 1,000 = 17,538,000 license plates.
Repeated Addition with Rational Numbers as Multiplicands
When the size of a set is not a whole number, using the unit fraction as the main unit leads to an easy process for
computing the product.
Example 16:
A small measuring cup has a capacity of 3/8 of a liter. How much water will two of these cups hold?
2 • 3/8 liter = 3 eighths of a liter + 3 eighths of a liter
= 6 eighths of a liter
= 6/8 L (or 3/4 L)
Example 17:
I bought three half-gallons of milk today. How many gallons of milk did I buy?
3 • 1/2 gallon = 1/2 gallon + 1/2 gallon + 1/2 gallon
= 3/2 gallons = 11/2 gallons
As these examples illustrate, we can find the product of a whole number and a rational number by multiplying the
number of unit fractions, i.e., the numerator: m • N = m • N
D
D
If a multiplication problem contains mixed numbers, change these mixed numbers to improper fractions to make use
of the above property.
Example 18:
It takes 12/3 yards of ribbon to make a bow. How much ribbon is needed for four bows?
4 • (12/3 yards) = 4 • 5 thirds of a yard = 20 thirds of a yard = 20/3 yd or 62/3 yards
Compare this to using feet as a unit: 4 • 5 thirds of a yard = 4 • 5 feet = 20 feet
5.1 Homework Problems
A. Answer the following.
1a. State the basic definition of multiplication.
b. In situations involving repeated addition, the total can be found by multiplying the ? of sets by the ? of a set.
2.
Define the following: (a) multiplicand; (b) multiplier; (c) row; (d) Cartesian product.
3a. List the four general situations leading to repeated addition.
b. Invent and solve your own example for each situation. Do not use the examples given in the text.
4.
Show how the area of a 3'' by 5'' rectangle can be found by repeated addition. Use a well-labeled diagram.
5.
Show how the number of elements in a 3 by 5 array can be found by repeated addition. Use a labeled diagram.
6.
Fill in the blanks.
(a) 4 • 3/5 = 4 • ? fifths = 12 ?
7a. Draw a picture to show why 2 • 3/5 = 6/5.
8.
b.
(b) 3 • 5/4 = 3 • 5 ? = 15 ?
Use repeated addition to find 2 • 3/5 = 6/5.
Explain why, in situations involving repeated addition, the multiplicand and the product have the same units.
Include an example.
9. Which of the following are arrays?
a. ♦ ♦ ♦ ♦
b.
1
45 48
♦ ♦ ♦
0
15
32
c.
☺☺☺☺☺☺
☺☺☺☺☺☺
d.
♣
♠
♠
♠
♣
♠
10. State the number of rows and columns and the total number of elements in each of the arrays in the previous
problem.
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11. Ron purchases three boxes of light bulbs. Each box contains 6 packages of bulbs, and each package contains
two bulbs. Find the total number of light bulbs purchased by using:
a. a series of repeated additions b. multiplication
c. a picture
d. a tree diagram
12. Use a tree diagram to find the number of different pizzas if there are three types of crusts (thin, medium, or
thick), two types of dough (white or whole wheat), and four kinds of topping combinations (plain, pepperoni,
super, and vegetarian).
13. The screen on a calculator contains pixels arranged in 62 columns and 48 rows. How many pixels occupy the
screen? (A pixel is a single position on the screen. It is either lighted or unlighted.) Draw the beginnings of an
array and solve this problem.
14. Ryan now has only 62 toy soldiers, after losing 48 in the woods yesterday.
a. How many toy soldiers did Ryan have before playing with them in the woods?
b. Identify the type of this problem.
15. An auditorium has 100 rows. The first row contains 20 chairs, and each succeeding row contains one more chair
than the previous row.
a. How many chairs are in the 100th row? Solve this problem by using an organized table containing at least three
rows and finding the pattern.
b. How many chairs are there altogether in the auditorium? [Hint: What is the sum of the chairs in the 1 st and
100th row? What is the sum of the chairs in the 2nd and 99th row?]
16.
a.
b.
c.
d.
License plates for a certain state contain 4 letters followed by 3 digits.
State one possible license plate for this state.
How many different license plates are possible?
How many license plates starting with LOVE are possible?
If 18 four-letter words are eliminated from the possible choices of four-letter combinations, and the use of “000”
is eliminated, how many different license plates are possible?
17. Some lottery tickets consist of six digits. What are your chances of winning the lottery if there is only one
winning combination of digits?
18. A large bag of mulch is labeled as containing 2 cubic feet of mulch. How many cubic inches of mulch is this?
[Hint: One cubic foot is 12'' by 12'' by 12''.]
19. A 10′ by 8′ patio is to be made with cement. It will be 2'' thick. How much cement is needed?
20. Explain how the area of a right triangle is related to the area of a rectangle with the same base and height.
Include a diagram.
21. A clay brick measures 8'' long, 4'' deep, and 3'' high. It is hollow in the middle, with sides and bottom that are 1''
thick. A cubic inch of clay weighs about two ounces. How heavy is this brick?
8 cm
22. Find the volume of the wedge to the right.
3 cm
15 cm
23. A 20' by 30' rectangular swimming pool is 3' 4'' deep at one end
and steadily increases to 8' deep at the other end 30' away.
How many gallons of water does it hold? (There are about
7½ gallons of water in one cubic foot.)
_______________________________________________
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_______________________________
_________________________________________________________
____________________
5.2 Division in the Context of Repeated Addition
Like multiplication, division is a derived operation. It is possible to solve many division problems by using more
basic operations, as illustrated in the next activity.
Activity 5.2A
A. Show how to solve the following problems using counting, addition, or subtraction. Use pictures or diagrams as
appropriate.
1.
A kindergarten teacher has one of her children distribute 10 lollipops equally to five children. The child gives
one to each child, then another, and another, until they are all gone. How many lollipops does each child get?
2.
A class contains 24 children seated at tables in groups of four. How many tables are there?
3.
I cut 3 apples in half and gave away all the half-apples, one to each child in the room. How many children are in
the room?
B. Travis, Zack, and Chad are playing with toy soldiers. Travis has eight toy soldiers, Zack has six, and Chad has
fourteen. All three boys organize their soldiers into pairs. Then Travis and Zack team up against Chad.
1.
Compare the pairs in each “army.” This situation illustrates that (8  2) + (6  2) is the same as (___ + __)  2.
2.
Make a generalization using fraction form: A + B =______________________________
C
C
_________________
A. The Basic Definition of Division
Just as subtraction is the inverse of addition, division is the inverse of multiplication.
BASIC DEFINITION OF DIVISION
Division is the Inverse of Multiplication.
A ÷ B =  is equivalent to B   = A, for B ≠ 0.
The first number in a division is called the dividend, the second is the divisor, and the result is the quotient.
Dividend ÷ Divisor = Quotient
Example 1:
Consider 12 ÷ 3 = 4.
12 is the dividend, 3 is the divisor, and 4 is the quotient.
12 ÷ 3 = 4 because 12 = 3 • 4.
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In other words, if we can formulate a problem into the multiplication sentence, A •  = C, then we can find the
unknown factor by reformulating the sentence into a division sentence:  = C ÷ A. Notice that the product in the
multiplication sentence corresponds to the dividend in the corresponding division sentence.
Example 2:
The floor of a right rectangular solid measures 3 m by 2 m, and the solid has a volume of 30 m 3.
What is the height of the solid?
V = LWH => 30 = 3 • 2 • H => 30 = 6 • H. So H = 30 m3 ÷ 6 m2 = 5 m.
B. Two Major Interpretations of Division
All situations involving division are equivalent to multiplication problems with a missing factor. However, two quite
different situations give rise to division.
1. Division as Partitioning:
Total ÷ Number of Parts = Size of the Part
The total is known, the number of sets (multiplier) is known, but the size of the set (multiplicand) is unknown.
Example 3:
Ten candies were distributed equally to five children. How many candies did each child get?
Solution A
The problem is to determine the size of the set given the number of sets. The solution can be found
by partitioning. Ten partitioned into five equal parts yields two candies per part.
§ §
Solution B
§ §
§ §
§ §
§ §
We have an unknown multiplicand, namely the number of candies given to each child. Thus we have
5 • B = 10. By the definition of division, B = 10 ÷ 5.
Teaching Tip: Young children can partition a set by dealing out the elements in the set like cards in a card game.
Later on, such experiences with partitioning can help children understand this basic meaning of division.
Example 4:
A pizza has been cut into eight equal pieces, and Anne eats two pieces. If two people share the
remaining pizza equally, how much of a pizza will each person eat?
If six pieces are split evenly between two people, each person will get three pieces.
As these examples illustrate, division can be used to find the size of a part given the original quantity and the number
of parts into which it is partitioned. This is called the partitioning interpretation of division.
Partitioning Interpretation of Division
For B a natural number, A ÷ m can be interpreted to mean
the size of a part when A is partitioned into m equal parts.
A
Am
m parts
Units in Partitioning Problems
In situations involving partitioning, the quotient is the size of a part when the dividend is partitioned into the number
286
of parts specified by the divisor. Hence the quotient, as part of the dividend, has the same unit as the dividend.
Example 5:
Sixty feet of rope is cut into 12 pieces of equal length. How long is each piece?
60 feet ÷ 12 = 5 feet
2. Division as Repeated Subtraction: Total ÷ Size of the Part = Number of Parts
Example 6:
A class contains 24 children seated at tables in groups of four. How many tables are there?
* *
* *
* *
* * ...
* *
* *
* *
* *
<---------- How many tables? ------------>
= 24
Solution A
Add fours until we reach 24: 4 + 4 = 8, 8 + 4 = 12, 12 + 4 = 16, 16 + 4 = 20, 20 + 4 = 24.
We added 6 fours to get 24, so the answer is 6 tables.
Solution B
Subtract 4 repeatedly from 24 until we reach 0: 24 - 4 - 4 – 4 - 4 - 4 - 4 = 0. We had to subtract
six fours, so there are six tables.
Solution C
Find a missing multiplier m so that m • 4 = 24. That is find m such that m = 24 ÷ 4.
Division as repeated subtraction occurs in situations where a known quantity has been partitioned into equal parts of
a known size. The problem is to determine the number of parts.
Repeated Subtraction Interpretation of Division
For B ≠ 0, A ÷ B can be interpreted to mean the number of B’s contained in A,
or the number of times B can be subtracted from A.
A
B
B
B
B
B
B
A÷B
Number of parts of size B in set A
Stated another way, we have: A
Example 7:
- B - B - B... - B = 0
Since 36 - 9 - 9 - 9 - 9 = 0, we have 36 ÷ 9 = 4.
Units in Repeated Subtraction
In situations involving repeated subtraction, the quotient is the number of divisors in the dividend. Hence the
quotient does not have a reference unit. For this reason we say that the units of the dividend and divisor “divide
out,” just as common factors divide out.
Example 8:
How many 200’s are in 600?
There are 3 sets of 200’s in 600. Thus we can say that in the division of 6 hundred by 2 hundred , the
hundreds units divide out.
Example 9:
A child arranges six toy soldiers into sets of two soldiers each. How many sets are there?
6 toy soldiers ÷ 2 toy soldiers = 3 => There are 3 sets of two soldiers in the set of six soldiers.
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C. Rational Numbers in Division
1. Quotients as Rational Numbers
Partitioning whole numbers can lead to parts with fractional sizes. Such problems reveal a surprising connection
between quotients and fractions.
Activity 5.2B
A. Three pizzas are to be shared equally among four people. How much pizza does each person get?
1.
The problem is to partition 3 pizzas into 4 equal parts and determine the size of the part.
That is, we want to find _____________ ÷ ___.
2a. Draw a diagram that shows how to solve this problem by cutting each pizza into four pieces. Shade the pieces to
be claimed by the first person.
b. We have 3 pizzas ÷ 4 = 12 _______ of a pizza ÷ 4 = 3 ___________.
3.
Thus 3 ÷ 4 is equivalent to the rational number _______.
B. Use diagrams to solve the following problems.
1.
Adrien’s will states that one ninth of his estate is to be given to a certain charity and the remaining part of the
estate is to be partitioned equally between his two children. What fraction of the estate will each child inherit?
2.
Anna’s will stated that one quarter of her estate was to be given to a certain charity and the remaining part of the
estate was to be partitioned equally between her two children. What fraction of the estate did each child inherit?
The above activity illustrates the following relationship between quotients and fractions.
The Connection Between Quotients and Fractions
For any real numbers A and B, with B ≠ 0, A ÷ B is the same as A/B.
The relationship between A/B and A  B is not obvious. For instance, consider 3 ÷ 5 and 3/5. We can interpret 3 ÷ 5
to mean the size of a part when three units are partitioned into five equal parts; we can interpret 3/5 to mean three of
five equal parts of one unit. On the face of it, these seem to be very different problems. They are certainly different
processes. Yet, as the following example illustrates, they yield the same result.
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Example 10:
To partition 3 acres into 5 equal parts:
1 acre
1 acre
1 acre
a. Convert 3 acres into 15 fifths of an acre.
b. 15 fifths of an acre ÷ 5 = 3 fifths of an acre = 3/5 acre
Thus we have three interpretations for a fraction A/B.
1.
A/B
2.
A/B
3.
A/B
can refer to A parts of a unit that has been partitioned into B equal parts.
Example: “3/5 of an acre” refers to three parts of an acre that has been partitioned into five equal parts.
can refer to the ratio of two quantities, where for every A elements in the first quantity there are B elements
in the second quantity.
Example: “The ratio of girls to boys in our class is 3/5” means that there are three girls for every five boys.
can refer to A divided by B. This interpretation has multiple meanings, including partitioning and repeated
subtraction.
Example: If three acres of land are to be shared equally by five heirs to an estate, then each heir receives
3 acres ÷ 5 or 3/5 of an acre.
2. Rational Number Dividends and Divisors
What is the meaning of an expression like 3/4 ÷ 2? This division of a fraction by a whole number can be interpreted
as partitioning. Just as with whole numbers, the key to partitioning a fraction into two equal parts is to convert the
fraction into a form that includes a multiple of two.
Example 11:
Partition 3/4 of a pizza equally between two people.
Cut each of the fourths into two parts. That is, convert 3/4 to 6/8. Now we have:
6 eighths of a pizza ÷ 2 = 3 eighths of a pizza = 3/8 pizza
What is the meaning of an expression like 3 ÷ 3/4 or 3/4 ÷1/8? These divisions can be interpreted in the context of
repeated subtractions as the next activity illustrates.
Activity 5.2C
A. Mary is going to make giant three-quarter-pound hamburgers. How many hamburgers can she make with three
pounds of hamburger meat?
1.
Solve this problem using repeated subtraction.
2.
The problem is to find out how many quarter-pounds are in 3 pounds.
a. The division associated with this problem is 3 lbs ÷ _____ lb.
b. Convert 3 lbs to quarter-pounds.
c. 3 lbs ÷ 3/4 lb = ___ quarter-pounds ÷ ___ quarter-pounds = _____ (Note that the units cancel out.)
d. So Mary can make ____ hamburgers.
B. Solve the following problems without using standard algorithms.
1.
If a strip of metal that is 15 sixteenths of an inch long is cut into pieces that are 3 sixteenths of an inch long, how
many of these smaller pieces will there be?
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2.
If a strip of metal that is 3/4 of a foot long is cut into pieces that are 1/8 of a foot long, how many of these smaller
pieces will there be?
3.
Suppose a restaurant serves a glass of wine that is 1/8 of a liter. If a box of wine holds 15/6 liters, how many
glasses of wine can be poured from this box? Include the fractional part of a glass in your answer. [Hint:
Convert to twenty-fourths.]
Understanding the process of dividing a fraction by a fraction is not straightforward. To make sense of these types
of division, it is helpful to use the repeated subtraction interpretation of division and a common unit. As the
following examples illustrate, this boils down to finding a common denominator.
Example 12:
Suppose six acres are divided into three-quarter-acre lots. How many lots will there be?
6 acres = 24 quarter-acres
Example 13:
=>
6 acres  3/4 acre = 24 quarter-acres  3 quarter-acres = 8
If 21/2 tons of gravel are to be poured into bins each holding half of a ton, how many bins are
needed?
Convert to half-tons: 21/2 tons  1/2 ton = 5 half-tons  1 half-ton = 5
Fortunately a relatively simple pattern occurs. Following is the explanation for this pattern.
1.
Use the Fundamental Property of Fractions to generate equivalent
fractions with the same denominator.
2.
Since AD and BC have the same unit, namely the unit fraction 1/BD,
this division can be interpreted to mean “How many BC’s are in AD?”
3.
As we shall see, a quotient can be interpreted as a fraction.
4.
The Shortcut
A/B
AD/BD
 C/D = AD/BD  BC/BD
 BC/BD = AD ÷ BC
AD ÷ BC = BC/BD
A/B
÷ C/D = AD/BC
Teaching Tip: Sometimes this shortcut is called “cross-multiplying.” This is a very bad idea. “Crossmultiplying” more commonly refers to a shortcut used to solve proportions. For instance, the proportion 3/x = 8/5
can be solved by “cross-multiplying” to obtain the equivalent equation 3 · 5 = 8x. In contrast, the result of “crossmultiplying” when dividing fractions is a fraction, not an equation. When different processes are referred to by the
same name, students often confuse the results. Thus it is better not to refer to the division of fractions as “crossmultiplying.” A pedagogically better way of computing the quotient of two fractions, which involves inverting the
divisor, will be discussed later in this chapter.
Example 14:
Finding 11/2  1/4 using a variety of methods:
(a) Repeated subtraction as visualization: In your mind’s eye, visualize the number of quarter
pieces of pizza in 11/2 pizzas. There are six such pieces.
(b) Formal repeated subtraction: 11/2 - 1/4 - 1/4 - 1/4 - 1/4 - 1/4 - 1/4 = 0 => 11/2  1/4 = 6
(c) Common unit: 11/2  1/4 = 6 fourths  1 fourth = 6
(d) Shortcut:
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11/2  1/4 = 3/2 ÷ 1/4 = (3 · 4)/(2 · 1) = 6
D. Remainders and Two Useful Theorems
It is a curious fact that inverse operations are often not as well behaved as the original operations. Here is a case in
point: multiplying two whole numbers yields a whole number, but dividing two whole numbers can result in a
remainder.
Activity 5.2D
1. It takes 15 inches of ribbon to make a certain kind of bow.
a. Suppose Mary has 50 inches of ribbon. How many bows can she make with this ribbon, and how much ribbon
will be left over?
b. Specify a length of ribbon that can be used to make bows without having any ribbon left over.
c. Give a general description of the lengths of ribbon that can be used to make bows without having any ribbon left
over.
d. Use your calculator to determine how much ribbon will be left over if Mary makes as many ribbons as possible
from a roll containing 88 feet of ribbon. Report your answer in inches.
2.
The maximum class size for kindergartners in one state is 18. A school has 50 kindergartners. What is the
smallest number of kindergarten classes that this school must have?
3.
At a practice, a coach divides his team into groups of four girls each. He assigns any remaining players to be
referees. If 23 players show up, how many will be referees?
4.
Three children steal into the kitchen late one night and find their mother’s secret cache of 11 chocolate bars.
a. If the children decide to split the chocolate bars evenly, how many chocolate bars
does each child get?
_____________
b. In the context of this problem, explain the meaning of the remainder of 2 in the equation 11  3 = 3 R 2.
c. Explain what happened to this whole number remainder in this problem.
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Division will lead to a “left-over” when the dividend is not a whole number multiple of the divisor.
Example 15:
Twenty-six grapefruits are being packed into boxes that hold six grapefruits each. How many boxes
will be filled, and how many grapefruits will be left over?
26 is not a multiple of 6. Instead, 26 = 4 • 6 + 2. So there will be four full boxes with two
grapefruits left over.
26 grapefruits
6 grapefruits
6 grapefruits
6 grapefruits
6 grapefruits
2 g.f.
In general, if A and B are whole numbers, then either (a) A is a whole-number multiple of B or (b) A is the sum of a
whole-number multiple of B and a remainder. The remainder has the same unit as the dividend. The relationship
between the dividend A, the divisor B, the whole number quotient q and the remainder r is summarized as follows.
The Division Theorem
For any whole numbers A and B, with B ≠ 0, A can be written as qB + r,
where q and r are unique whole numbers, with 0 ≤ r < B.
A
q B’s
r
This theorem is called the Division Theorem because of the connection between A divided by B and A written as
q · B + r. If A consists of q B’s plus a remainder r, then A ÷ B is equal to q with a remainder of r.
Example 16:
The following statements convey the same information:
a. 242 = 5 • 43 + 27
b. 242 contains 5 forty-threes with 27 left over.
c. 242 ÷ 43 is equal to 5 with a remainder of 27.
It is common (at least in elementary school) to indicate a whole-number quotient and remainder using the “R”
notation, as illustrated in the next example. Note that “R” does not indicate addition.
Example 17:
“14 ÷ 5 = 2 R 4” means that 14 = (2 • 5) + 4. In other words, 14 contains 2 fives with 4 left over.
Another useful theorem related to division is illustrated in the following example.
Example 18:
Bridge is a card game involving exactly four players. Marge is organizing a bridge party at her
retirement community. First eight people sign up, so Marge prepares two tables for four. Then
another 12 people sign up, so Marge prepares three more tables for a total of five tables. Obviously,
if all 20 people had signed up at the same time, Marge would also have prepared five tables. This
illustrates the following fact: 20 = 12 + 8 = 12 + 8
4
4
4
4
In general we have the following result.
Quotient of a Sum Property
If A, B, and C are real numbers with C  0, then A + B = A + B
C
C
C
This is called the Quotient of a Sum Property because it states that the quotient of a sum (A + B) is the same as the
sum of the quotients A/C and B/C.
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Teaching Tip: Many students find the Quotient of a Sum Property rather strange when it is read from left to right.
Just ask them to read the property from right to left—in this direction the property should be very familiar!
See how the Quotient of a Sum Property plays a role in the next example.
Example 19:
Forty-one acres are to be divided into eight lots of equal size. What will be the size of each lot?
Since 41 acres = 8 · 5 acres + 1 acre, each lot will include 5 acres. If the remaining acre is
partitioned equally among the eight lots, each lot will increase by an eighth of an acre. Thus the total
size of each lot will be 51/8 acres.
Summary: 41 acres/8 = 40 acres/8 + 1 acre/8 = 5 acres + 1/8 acre = 51/8 acres
As this example shows, a quotient can be expressed as a non-whole number that includes the remainder as a
fractional part of the divisor.
If A = qB + r, then A  B = qB + R = qB + r = q + r
B
Example 20:
B
B
B
387  8 = (48 · 8 + 3)  8 = 48· 8 + 3 = 48 · 8 + 3 = 48 + 3 = 48⅜
8
8
8
8
The concept of whole number quotients also applies to problems involving fractional dividends and divisors. In such
cases, be careful to interpret the remainder correctly.
Example 21:
Suppose three and a quarter liters of acid is being poured into half-liter containers.
a. How many containers will be filled? Include fractional parts.
Compute the answer using the shortcut: 31/4 liters  1/2 liters = 13/4  2/1 = 13/2 = 61/2
This means that 61/2 containers will be filled.
b. How many full containers will there be, and how much acid will be left over?
Since 31/4  1/2 = 61/2, there will be six full containers. The left-over acid would fill 1/2 of a half
liter container, so there is 1/4 of a liter of left-over acid.
Remember that the fractional part of a quotient is equal to the remainder divided by the divisor. To find the
remainder in terms of original units, multiply the fractional part of the quotient by the divisor.
Finding Whole Number Remainders from Quotients in Decimal Form
If a calculator is used to find a quotient, the answer is usually expressed in decimal form. The whole number
quotient q is clearly identifiable as the whole number part of this decimal. One way to find the whole number
remainder is to use the relationship between A, B, q, and r: A = qB + r. Solving this for r yields the following
equation: r = A – qB. In other words, find r by subtracting q B’s from A.
Example 22:
242 ÷ 43 = 5.6279069…
=>
242 = 5 · 43 + r
=>
r = 242 – 5 · 43 = 27
Described in another way: When we compute 242 ÷ 43 as 5.62…, we have determined that there are
five 43’s in 242, plus a remainder. To find the remainder, subtract the five 43’s from 242.
Another way to find the whole number remainder r is to recognize that the fractional part of the decimal represents
the ratio of r to the divisor. Thus r can be found by multiplying this fractional part by the divisor. Avoid rounding
errors by using all the digits provided by your calculator for the fractional part.
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242 ÷ 43 = 5.6279069…
Example 23:
=> r = 43 · 0.6278069… = 27
Situations Involving Whole Number Quotients and Remainders
While there are many division situations in which the answer is a non-whole number quotient, there are many
division situations in which the answer must be a whole number. These situations usually involve units that are
indivisible, i.e., units that cannot be partitioned into smaller units.
The organizer of the school’s May Day event decides to form six rows of chairs for the audience.
She wants the same number of chairs in each row. There are eighty-seven chairs available. How
many chairs should be in each row?
Example 24:
Find 87 ÷ 6 = 14 r 3. This means that 87 = 14 • 6 + 3. Put 14 chairs in each row, with three chairs
left over.
The sixth grade is scheduled to see the play “The Lion King,” but the bus has broken down. Parents
with minivans are being recruited to take all 87 sixth graders to the play. If each minivan carries six
passengers (not including the driver), how many parents with minivans need to be recruited?
Example 25:
Since 87 = 14 • 6 + 3, we can fill up 14 vans and part of another van. This means we need 15 vans to
take all 87 sixth graders to the play. (Alternately, line up 14 parents with minivans and one parent
with a sedan.)
As the above examples illustrate, sometimes the quotient is rounded up and sometimes it is rounded down to find the
appropriate answer to a question. Use common sense to decide which way to round.
Sometimes the remainder plays the starring role in a division problem! That is, sometimes the relevant part of a
division is not the quotient but the remainder. Consider the next examples.
Example 26:
January 1, 2002 fell on a Tuesday. On what day did January 31, 2005 fall?
Starting with January 1, every seven days there will be another Tuesday. January 29 will fall on a
Tuesday because it is 28 days after January 1. Thus January 31 will fall on a Thursday.
Example 27:
December 25, 2005 falls on a Sunday. On what day will December 25, 2009 fall?
There are 365 days in most years and 365 = 52 • 7 + 1. This means that a year consists of 52 full
weeks plus a day. That extra day, the remainder in the division 365  7, means that from one 365-day
year to the next, every date moves forward one day. So December 25, 2006 will fall on a Monday
and December 25, 2007, will fall on a Tuesday. The year 2008 is a leap year with 366 days, the
extra day occurring on February 29. This means that all dates after February 29 move forward two
days from the previous year. Thus December 25, 2008 will fall on Thursday. December 25, 2009,
will fall on a Friday.
Teaching Tip: An efficient way to identify leap years, which normally occur when the year is divisible by four, is to
use the following property: a whole number is divisible by four if and only if the last two digits are divisible by four.
For example, 2036 will be a leap year because 36 is divisible by 4.
Various examples in this section have illustrated four effects of the remainder. These are summarized below.
Four Possible Effects of the Remainder
1.
2.
3.
4.
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Eliminate the remainder. Round the quotient down to the nearest whole number.
Round the quotient up to the next whole number.
Retain the remainder as the answer.
Include the remainder in the answer as a fractional part of the divisor.
Teaching Tip: Students have been known to lose track of the existence of whole number quotients and remainders
in later grades because they become so accustomed to using calculators that yield only decimal quotients. Their
memories can be jogged by working problems that require whole number answers, not decimal answers.
Summary
Division is defined as the inverse of multiplication. From an understanding of multiplication as finding a total given
a number of repeated sets, there arise two understandings of division. The first is to find the size of the repeated set.
The second is to determine the number of these repeated sets. Complications occur because of the backwards nature
of division, especially as it relates to the existence of remainders and the behavior of rational numbers.
5.2 Homework Problems
A. Concepts
1.
a.
b.
c.
Definitions, Properties, and Vocabulary
State the basic definition of division.
Use the basic definition of division to rewrite A  ⅜ =  as a multiplication sentence.
Rewrite the following multiplication sentence as a division sentence: 4   = 2/3
2a. Use the basic definition of division to rewrite 8  0 =  as a multiplication sentence.
b. Explain why this multiplication sentence, and hence the division sentence, has no solution.
3.
Identify the divisor, dividend, and quotient in the following division sentence: 6  1/3 = 18.
4.
a.
b.
c.
List three numbers in each of the following sets.
Multiples of 12
Factors of 12
Numbers divisible by 12
5. Justify your answers to the following.
a. Is 24 a multiple of 8?
b. Is 24 divisible by 8?
d. Is 0 a multiple of 8?
e. Is 0 divisible by 8?
6.
c. Is 24 a factor of 8?
f. Is 0 a factor of 8?
Why can division always be interpreted as the process of finding an unknown factor?
7. Which of the following can be interpreted as A  B, for B  0?
a. A/B
b. A : B
c. Number of B’s in A
d. , where A   = B
8.
a.
b.
c.
Explain the meaning of 5/6 using:
the basic definition of an elementary fraction;
division interpreted as partitioning;
division interpreted as repeated subtraction, with a whole number quotient and remainder.
9.
a.
b.
c.
d.
e.
The Division Theorem
For any two whole numbers A and B, A can be written as a ? of B’s plus a ? .
Show this relationship for A = 17 and B = 3.
Show this relationship for A = 6 and B = 17.
If A = cB + d, describe A  B.
Fill in the blanks: 37,893 = ?  87 + ? and 37,893  87 = ? R ?
10. Fill in the blanks.
a. If 27 ÷ 4 = 63/4, then 27 = ? • 4 + ?.
b. If 473 = 8 • 56 + 25, then 473 ÷ ? = 8 + 25/?.
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11. Which of the following are equivalent to 56 = 9 • 6 + 2?
a. 56 ÷ 9 = 6 R 2
b. 56 ÷ 6 = 9 R 2
c. 56 ÷ 9 = 62/56
d. 56 ÷ 9 = 62/9
e. 56 ÷ 6 = 9 + 2
12.
a.
b.
c.
The Quotient of a Sum Theorem
State the sum that is the same as (x + y)/z.
According to the Quotient of a Sum Theorem, 96/3 is the same as 90/3 + ? .
Determining the number of threes in 96 is the same as determining the number of threes in 90 and adding this to
the number of threes in ? .
d. The Quotient of a Sum Theorem states that first adding A and B and then dividing the sum by C is the same as
first dividing A by C and dividing B by C and then ?.
B. Division as Partitioning
1.
Describe the meaning of 6  2 in terms of partitioning.
2.
Identify which of the following three quantities is unknown in a partitioning problem:
total, size of repeated set, number of repeated sets.
3.
a.
b.
c.
d.
Write and solve a story problem that involves partitioning for each of the following conditions:
The dividend is three fifths.
The quotient is three fifths.
The dividend is 0.
The divisor is 0.
4a. Identify which of the following three quantities have the same units in a partitioning problem: total, size of
repeated set, number of repeated sets.
b. Explain why these two quantities have the same unit. Include an example.
Use the partitioning interpretation of division to explain why A  A = 1, for A  0.
5.
6a. For division interpreted as partitioning: (total) ÷ (number of parts) = ? .
b. What type of number occurs as the divisor in a partitioning problem, and why?
7a. A ÷ B can be interpreted as the process of partitioning a set of size A into B parts and finding ? .
b. Using this interpretation, we have 8 people ÷ 2 = ? . Justify your answer.
C. Division as Repeated Subtraction
1.
Describe the meaning of 6  2 in terms of repeated subtraction.
2.
Identify which of the following three quantities is unknown in a repeated subtraction problem:
total, size of repeated set, number of repeated sets.
3.
a.
c.
e.
Write and solve a story problem that involves repeated subtraction for each of the following conditions:
The dividend is three fifths.
b. The quotient is three.
The dividend is 0.
d. The divisor is 0.
The divisor is 1/3.
4a. Identify which of the following three quantities have the same units in a repeated subtraction problem: total,
size of repeated set, number of repeated sets.
b. Explain why these two quantities have the same unit. Include a word problem as an illustration.
Use the repeated subtraction interpretation of division to explain why A  A = 1, for A  0.
5.
6a. A ÷ B can be interpreted as the process of finding how many times B must be subtracted from A to get ? .
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b. Using this interpretation, we have 6 feet ÷ 3 feet = ? because ? .
7a. Use the repeated subtraction interpretation of division to explain why 8 tenths ÷ 2 tenths = 4.
b. Explain why A/B  C/B = A  C in terms of repeated subtraction and the common unit of the dividend and
divisor.
8. Invent a story for each of the following and find the answers.
a. 18 lbs ÷ 3 lbs = ?
b. 18 lbs ÷ 3 = ?
9.
a.
b.
c.
d.
Which of the following can be computed by determining M  2?
What number should I multiply 2 by to get M?
What is the size of a part if M is partitioned into two parts of equal size?
How many twos are in M?
If M is partitioned into halves, how many halves will there be?
D. Rational Numbers and Division
1. Rational Divisors
a. Invent a story that can be solved by finding 31/3 ÷ 2/3.
b. Draw a labeled diagram that illustrates how to find the solution.
2. Rational Dividends
a. Invent a story that can be solved by finding 41/2 ÷ 3.
b. Draw a labeled diagram that illustrates how to find the solution.
3.
Explain why 15/8 ÷ 3/8 is the same as 15 ÷ 3 using the repeated subtraction interpretation of division and unit
fractions.
4.
a.
b.
c.
Rational Quotients
Use a diagram to illustrate how to divide two pizzas evenly among three people.
Fill in the blanks with appropriate unit fractions: 5 ÷ 6 = 30 ? ÷ 6 = 5 ?
Suppose 4 units are partitioned into M equal parts. Describe the size of a part.
5.
a.
b.
c.
Find 11/2 ÷ 3/8 by the following methods.
repeated subtraction
common denominators
a third method of your own choosing
6.
a.
b.
c.
d.
Which of the following can be computed by determining M  1/2?
What number should I multiply 1/2 by to get M?
What is the size of a part if M is partitioned into two parts of equal size?
How many twos are in M?
If M is partitioned into halves, how many halves will there be?
E. Remainders
1.
a.
b.
c.
Basics
Under what circumstances will division of whole numbers include a nonzero remainder?
When the remainder is 0, the dividend must be a (multiple/factor/term/product) of the divisor.
A remainder in a division problem can be considered as a fractional part of the ? .
2.
Find the whole number quotient and remainder for the division 4379  35.
3a. List the four possible effects of a remainder on the answer of a division problem.
b. Invent a word problem for each of these four effects.
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F.
Problem Solving
1.
The teacher decides to organize his class of 22 students into teams of four children each, with the “leftover”
children working with her. How many teams will there be, and how many children will be working with the
teacher?
2.
If a 73/5 acre lot is to be divided equally into 6 lots, what will be the size of each lot?
3.
I cut oranges into fourths and gave a piece to each of 22 children. How many whole oranges did I use?
4.
Twenty-five children are going on a field trip in vans holding 7 children each. How many vans are needed?
5.
January 1, 2004 falls on a Thursday. Determine the day of the week for January 1, 2012.
6.
The 15th day of a certain year falls on a Thursday. On what day of the week will the 327th day of the year fall?
7.
A construction company is paving a 21/4 mile stretch of freeway at the rate of 200 yards a day. How long will it
take to complete the job?
8.
The Martian year is almost exactly 687 days. Suppose Martians have seven-day weeks like we do. If the
Martian year of 2005 started on a Monday, on what day of the week would the Martian year of 2006 fall?
9.
On Venus the year is a little over 224 days. Suppose Venutians have five-day weeks (Monday through Friday),
with leap years that occur every three years and contain two extra days. The Venutian year of 2005 started on a
Monday and is a leap year.
a. On what day of the week will the Venutian year of 2006 start?
b. On what day of the week will the Venutian year of 2009 start?
10. The water in a tank weighs 668.75 pounds. One cubic foot of water weights 62.5 pounds. How many cubic feet
of water does the tank hold?
11. A manufacturer had a roll of 750 yards of linen goods that he cut into pieces 27 inches long to make dish towels.
He sold the towels at $4.80 a dozen.
a. If he sold all the towels, what was his revenue? [Hint: Revenue is the amount of money taken in.]
b. If the cost of producing and cutting the roll of linen goods was $380, what was the profit per towel?
12. A chemistry professor is preparing for a lab with 18 students. Each pair of students will need a tenth of a liter
of a 40% nitric acid solution for the day’s experiment. How much of this acid must the professor prepare?
13. A 31/4 yard strip of steel is to be used to make pieces that are a half foot long. How many pieces can be made,
and how much steel will be left over?
14. An estate worth one and a half million dollars is to be shared equally among five heirs. How much does each
heir inherit?
15. Eight and two thirds miles of interstate are to be paved in 20 days. How much road should be paved each day,
on average? Report your answer in feet.
16. How many nails weighing 3/8 of an ounce can be made from a third of a pound of metal?
____________________________________________________________________________________________
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______________________________________________________________________________________ ______
5.3 Multiplication as a Means of Comparison
Besides repeated addition, multiplication has a second major meaning. This is illustrated in the following activity.
Activity 5.3A
A. Jerry, Nick, and Melissa went running one Saturday morning. Melissa ran twice as far as Jerry. Nick ran two
thirds as far as Jerry. Let Jerry’s, Nick’s and Melissa’s distances be represented by J, N, and M respectively.
1.
Write an equation expressing the relationship between J and M.
______________
2. Suppose Jerry ran 12 miles.
a. Use a diagram to determine how far Nick ran.
b. Write an equation expressing the relationship between J and N.
3.
______________
In the last thirty years, there has been a 200% increase in the price of bread.
a. ____________________________________ is 200% of ___________________________________________
b. Label three sets in the following diagram: the old price,
the increase, and the new price.
c. If a loaf of bread cost 50¢ thirty years ago, how much does it cost now? Label the diagram
appropriately to find the answer.
4.
_____________
Suppose an employee gets one tenth off the sticker price.
a. ____________________________________ is 1/10 of _____________________________________________
b. How much will an employee pay for an item with a sticker price of $60? Label the following diagram using the
information in this problem and figure out the answer. Label three sets: the sticker price, the discount, and the
discounted price.
Multiplication in Comparison Situations
In the above problems, multiplication is used to describe the relationship between two quantities. In such situations,
the product is not a total but an amount that is described relative to a base of comparison. The multiplier indicates
how many or how much of the base is necessary to generate the described amount.
Described Amount = m • Base of Comparison
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Example 1:
Melissa ran twice as far as Jerry.
Let J = Jerry’s distance and M = Melissa’s distance:
M •______________•_______________•
We have M = 2 • J.
Example 2:
J •______________•
A 200% increase means that the increase is two times the original price. If the original price was
50¢, then the increase is 2 • 50¢, or 100¢. The new price will be 50¢ + 100¢ or $1.50.
Teaching Tip: Especially when an increase is over 100% of the original value, students may forget to add the
increase to the original price to find the final value. Warn them to be extra careful when they are working with
these types of problems.
Rational Number Multipliers
If the multiplier is a whole number, multiplication in comparison situations is similar to repeated addition. In the
above example, for instance, 2 • J still means J + J. Unlike repeated addition, however, multipliers in comparison
situations can be non-whole rational numbers. As the next example illustrates, the meaning of these multipliers is
directly based on the meaning of elementary fractions.
Example 3:
Nick ran two thirds as far as Jerry. This means that Nick’s distance N is two thirds of Jerry’s
distance J, or two of three equal parts of Jerry’s distance.
J •_____•_____•____•
N = 2/3 of J
N
_____•_____•
•
Since 2/3 plays exactly the same role in this example as 2, the multiplier 2 did in the previous example, it seems
reasonable to interpret “2/3 of J” as multiplication. For instance, if Jerry ran 12 miles, then 2/3 · J means to partition
12 into three equal parts and select two of these equal parts: 2/3 · 12 = (12 ÷ 3) · 2 = 8.
In general, for any positive rational number N/D, N/D · B means N/D of B, where N/D is interpreted as an elementary
fraction. That is, N/D · B means N of D equal parts of B: N/D · B = (B ÷ D) · N.
Example 4:
If Y = 7% · X, then Y is 7/100 of X, or seven of one hundred equal parts of X.
Example 5:
The guests ate two thirds of a box of 24 candies. How many candies did they eat?
Solution A
To find 2/3 of 24, first partition 24 into three equal parts. This yields 8 candies in each part, with 16
candies in two parts. The guests ate 16 candies.
Solution B
2/3
· 24 candies = 2/3 of 24 candies = 2 · (24 candies ÷ 3) = 2 · 8 candies = 16 candies
As the next activity illustrates, this process does not always yield a whole number.
Activity 5.3B
A. John ordered eight pizzas for a party. His guests ate two-thirds of all the pizza. How much pizza did they eat?
1.
Solve this problem by partitioning the pizzas below and shading the pizzas that were eaten.
2.
2/3
300
of 8 pizzas = 2/3 of ____ thirds of a pizza = 16 ___________________________ = 5 1/3 ____________
B. Four fifths of the City Council of Newton voted in favor of a tax increase. Of those in favor of a tax increase,
two thirds indicated that the increase should be more than 1%. What fraction of the city council was in favor of
a tax increase over 1%?
1.
Suppose the large rectangle to the right represents the Newton City Council.
a. Shade the area representing those who voted in favor of a tax increase.
b. Stripe the area representing those who favored an increase of more than 1%.
c. Use this diagram to find the answer to the question.
2.
______________
Symbolically:
(1) The problem is to find _____ of _____ of the city council.
(2) Convert the base so that its numerator is a multiple of 3: 4/5 = 12/____.
3a. Solve the following problem by using fifteenths as the unit.
2/3 · 4/5
= 2/3 of 12/15 = 2/3 of 12 _______________ = 8 ________________ or 8/____
b. The pattern that occurs indicates the following shortcut: 2/3 · 4/5 = (2 · 4)/(___ · ___)
Teaching Tip: Fractions such as 4/5 can be written as either “four-fifths” or “four fifths.” The use of two separate
words emphasizes “fifths” as the primary unit; the use of a hyphenated word emphasizes 4/5 as a single unit.
Parts of Parts
As the last problem in the above activity illustrates, it is common to describe parts of parts using multiplicative
comparisons. This leads to expressions such as “2/3 of 4/5 of the City Council.” How much is 2/3 of 4/5? The
following example shows several ways of determining the answer, all involving the identification of fifteenths as the
key unit.
Example 6:
Four fifths of the class passed the test. Of those who passed, two thirds made at least a B. What
fraction of the class made at least a B?
Students making at least a B = 2/3 of those who passed
= 2/3 of 4/5 of the class
= (2/3 · 4/5) of the class
Solution A
Use the Fundamental Property of Fractions to convert 4/5 to an equivalent fraction with a numerator
that is a multiple of three: 2/3 · 4/5 = 2/3 of 4/5 = 2/3 of 12/15 = 2/3 of 12 fifteenths = 8 fifteenths.
Solution B
Use a one-dimensional line segment partitioned into five equal parts. Partition each of these parts
into three parts and identify 2/3 of the small parts within 4/5 of class.
4/5 of class
_
__
_
|
| __|__ _|_ __|
4/5 = 12/15
|
|_._ _|_ _ _|_ _ _|_ _ _|
|
2/3 of 12/15 = 8/15
Solution C
Use a two-dimensional area diagram. Use vertical lines to partition the rectangle into five equal parts
and then use horizontal lines to partition 4/5 into thirds. Extend the horizontal lines to partition the
entire rectangle into thirds in order to determine the size of the smallest part relative to the whole.
301
4/5
of the whole
the whole
2/3
of 4/5 of the whole = 8/15 of the whole
Partitioning a quantity into five parts and then partitioning each of these five parts into three parts
creates a total of 15 parts. As the diagram illustrates, 2/3 ·of 4/5 includes 8 of these 15 parts, or 8/15.
The above example indicates that there is a surprisingly simple way to compute the product of two fractions: simply
multiply the numerators and multiply the denominators: A . C = A · C.
B D B·D
Thus, for example, we can compute 2/3 · 4/5 as follows: 2/3 · 4/5 = (2 · 4)/(3 · 5) = 8/15. The justification for this easy
shortcut is not the least bit obvious. The two-dimensional area diagram may be the most straightforward way to
verify that this shortcut works, but the most fundamental explanation is based on understanding that 2/3 · 4/5 means
2/3 of 4/5, and recognizing that 2/3 of 4/5 is the same as 2/3 of 12 fifteenths.
Teaching Tip: A good algorithm for computing the quotient of rational numbers can be obtained by combining
two patterns. We have just noted that A/B • D/C = AD/BC. Previously we found that A/B ÷ C/D = AD/BC. So we have:
A ÷ C = A . D = AD
B
D
B C
BC
Since D/C is the inverse of C/D, this rule can be summarized as follows: “To divide fractions , invert the second
fraction and multiply.” Be sure to stress to students that the second fraction, not the first, is to be inverted.
Multiplication with Decimals and Percents
If the multiplier m is between 0 and 1, m is often expressed
in percent form. While the form of the multiplier has no effect
on the meaning of the comparison, the use of percent (which
means hundredths) as a unit makes the use of grid paper
almost a necessity for drawing an illustrative diagram.
Example 7:
A is 3/4 of B => A = 3/4 · B
=> A = 75% · B
To compute answers, convert percents to decimal form and use the rules for decimal multiplication. (Justifications
for these rules will be discussed later.)
Example 8:
Becky invested 60% of her bonus in bonds and put the rest in her savings account. If her bonus was
$2500, how much money did she put in her savings account?
Amount invested in bonds = 60% of B, where B is the bonus
=> Amount left in savings = 40% of bonus
= 0.4 · $2500
= $1000
B
bonds
60% of B
savings
40% of B
Identifying the Components of Multiplicative Comparisons
To understand a multiplicative comparison, it is very important to identify the described amount and the base of
comparison. As the next activity illustrates, this is not as easy to do as one might think.
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Activity 5.3C
1.
State the amount being referred to by the number in the following situations.
a. Alexandria’s salary now is three times what it was at her part-time position.
__________________________
b. One-third of my salary is used to pay my rent.
___________________________
c. Hamilton County has a 9.25% sales tax.
___________________________
2.
For each of the above situations, describe the base to which the described amount is being compared.
a. ___________________________ b. ___________________________
3.
c. ___________________________
Suppose a real estate agent earns a 10% commission for selling a house. Fill in the following blanks.
______________________________________ is 10% of __________________________________________
4.
Suppose you buy an item at a 1/4 off sale. Fill in the following boxes and blanks with either “original price,”
“sale price,” or “discount.”
a.
= 1/4 · _____________________
b. ________________
c. _________________ = 3/4 · ______________________
5. The newspaper reported that the price of gasoline jumped 9% from August 1 to August 2.
a. Identify each of the three amounts, F, G, and H in the following diagram as either “price on August 1,” “price on
August 2,” or “price increase.”
F
F
_______________________
G
G _______________________
H
H
__________________________
b. Fill in the following blanks with either “price on August 1,” “price on August 2,” “price increase,” or an
appropriate percent.
(1)
is 100% of F.
(2)
is 9% of
(3)
is ___________ % of ____________________________________
Here are some pointers for identifying the components of a multiplicative relationship
1.
Described Amount is (___)% of Base of Comparison
=>
A=m•B
A multiplicative relationship can always be phrased in the above form, which corresponds directly to the
equation A = m • B.
Example 9:
Gary’s commission is one tenth of the selling price.
=> commission = 1/10 · selling price
Selling Price
C
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All Students
Example 10:
Forty percent of the students are women.
=> The number of women is 40% of the students.
=> number of women = 40% of the students
Women Students
“% (Amount)” or “(Amount) rate”
2.
In many situations, the described amount is stated before or after the multiplier, with the multiplier expressed in
percent form. The base, often unspecified, is usually a total or the original amount.
The state has an 8% sales tax. If the sticker price is $30, how much is the tax?
Sales tax = 8% of sticker price = 0.08 · $30 = $2.40
Example 12:
The store gives a 15% employee discount.
employee discount = 15% · original price
3.
Example 11:
Part-Whole: Part = m · Whole
a. Described Part
A part of a set is often described relative to the size of the set (the whole).
Example 13:
One fourth of 40 students were sick. How many students were sick?
Number of sick students = 1/4 of total number of students
= 1/4 of 40
= 10
10
10
10
10
Total Number of Students
It is particularly common to describe a decrease relative to the original amount.
Decrease
Example 14:
The size of the class decreased by a third when the instructor
enforced the prerequisites.
Decrease = 1/3 of Original
Remaining Students
Original Class
It is common to describe decreases using percents without stating the base of comparison. The original amount
is always the base of comparison for a percent decrease.
Example 15:
“An 8% decrease in the price of gasoline” means that the decrease is 8% of the old price.
b. The Other Part
With the part-whole model, we get “two for the price of one.” For example, if we know that 1/4 of the students
are sick, then we also know that (1 - 1/4) or 3/4 of the students are not sick. If the multiplier is in percent form,
we find the multiplier for the other part by subtracting from 100%. (100% is equal to 1.)
Describing the Other Part of a Set
If A = 25% of B, then the other part = 75% of B.
A
Other Part
25% of B
75% of B
100% of B
304
Example 16:
At a 25% off sale, what is the sale price of an item originally priced at $34.95?
Let P represent the original price. Note that P is 100% of itself.
Sale price = Original Price - Discount
= 100% of P - 25% of P
= 75% of P
25% · P
= 0.75 · $34.95
75% · P
= $26.21
5.
100% P
Expanding Amounts
a. The Increase
In a situation in which the size of a set increases, the increase is often described relative to the original amount.
Example 17:
The value of a stock increases by 150%. If it used to be worth $6 a share, how much was the
increase and how much is the stock worth now?
Increase = 150% of old value
= 1.5 • $6.00
= $9.00
old value
New Value = $6 + $9 = $15
increase
New Value
The original amount is always the base of comparison for a percent increase.
Teaching Tip: Some students are disconcerted by the possibility that a percent may be larger than 100%. This
may be due to associating percents exclusively with the part-whole type of comparison. When a part is
compared to a whole, the percent certainly cannot exceed 100%. However, there are many types of
comparisons in which the described amount can be larger than the base of comparison. For instance, an
increase can exceed the original amount. In these situations the multiplier is larger than 100%.
b. The New Amount
We also get “two for the price of one” in increase situations because the new amount is the union of the old
amount and the increase. This means that the new amount can be described in terms of the old amount by
adding the percent increase to 100%.
The Relationship Between the New Amount N and the Original Amount B
B
100% of B
Increase
X% of B
New Amount
N = (100% + X%) of B
Example 18:
Tuition has increased by 15%. If the tuition was $4000, what is the new tuition?
Tuition increase = 15% • old tuition (T)
New Tuition
=
=
=
=
=
old tuition + increase
100% · T + 15% · T
115% · T
1.15 · $4000
$4600
Old Tuition
Increase
100% T
15% T
115% T
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Reporting Sensible Answers
There are some situations in which non-whole numbers do not make sense as answers. In such situations, round the
answer to the nearest whole number.
Example 19:
A teacher reported that two thirds of her class had done well on the year-end standardized tests. This
teacher has 25 students. How many of her students did well on the tests?
Number of students who did well = 2/3 of 25 = 16.666...
About 17 students did well on the tests.
5.3 Homework Problems
A. Basic Concepts
1. Invent a word problem for the expression “3  6” based on the following meanings of multiplication.
a. Repeated addition
b. Means of comparison
2.
a.
b.
c.
Consider the expressions “2/3  6” and “2/3 of 6.”
What is the relationship between these two expressions?
Explain the meaning of “2/3 of 6.” Include a labeled diagram.
Invent and solve a comparison word problem that is solved by computing 2/3  6.
3.
a.
b.
c.
d.
e.
f.
g.
h.
Which of the following are true in situations involving multiplicative comparisons?
The described amount is never more than the base of comparison.
The described amount must be a part of the base of comparison.
The described amount can be a whole number multiple of the base of comparison.
If one part of a set is 10% of the set, then the other part must be 90% of the set.
If a set increases in size by 10%, then the original set is 90% of the enlarged set.
If a set decreases in size by 10%, then the shrunken set is 90% of the original set.
In comparison situations, the amount is always described explicitly.
In comparison situations, the base of comparison is always described explicitly.
4.
a.
b.
c.
Fill in the blanks.
If A is 2/3 of B and B is 1/4 of C, then A is ? of C.
If A is 20% of B and B is 150% of C, then A is ? % of C.
If A = 0.4 · B and B = 0.8 · C, then A is ? · C.
5.
In the following diagrams, the base of comparison B is represented by heavily outlined rectangles. Describe the
shaded area in terms of B in each of these situations.
a.
b.
c.
d.
66⅔%B
6.
In the following diagrams there are two bases of comparison, B 1 and B2. B2 is represented by the largest
rectangle and B1 is represented by the next largest rectangle. A is represented by the shaded area. For each
diagram, estimate the multiplier for the following multiplicative relationships: (a) A and B 1; (b) B1 and B2;
(c) A and B2. [Hint: Extend lines and draw extra lines to make good estimates.]
Example:
306
(a) A is 1/2 of B1 (B1 is striped.)
(b) B1 is 1/3 of B2.
(c) A is 1/6 of B2.
a.
7.
a.
c.
e.
b.
Suppose Y has the following length:
a length that is twice the length of Y
a length that is one fourth the length of Y
a length that is 50% more than Y
c.
d.
. If possible, accurately draw the following lengths.
b. a length that is 2 units longer than Y
d. a length that is a fourth of a unit less than Y
f. a length that is 25% less than Y
8a. Explain the meaning of 3/5 of a number M without making reference to multiplication.
b. What is the meaning of A/B • M, where A/B is a positive rational number?
c. A/B • 23 can be computed by dividing 23 by ? and multiply the result by ? .
9.
Explain why 1/5 of 3 is the same as 3 ÷ 5, with the latter interpreted as partitioning.
10. Which of the following are equivalent to 3/5 • B?
a. 3 of 5 equal parts of B
b. 3 • (B ÷ 5)
c. B ÷ 3/5
d. Partitioning B into 5 equal parts and selecting three parts
11. Find the following products of rational numbers using unit fractions and the definition of elementary fractions.
a. 2 • 6/5 = 2 • ? fifths = ? fifths
b. 1/3 of 7 feet = 1/3 of 21 ? of a foot = ?
1/
10/
1/
c. 5 • 11 = 5 of ___ elevenths = ?
d. 1/6 • 5/3 = 1/6 of 30 ? = ?
12. Develop examples to show that “of” does not necessarily mean “times,” while “times” usually means “of.”
13.
a.
b.
c.
Use each of the following methods to find 1/4 • 1/3.
Creating an equivalent fraction with a numerator that is a multiple of 4
Partitioning a one-dimensional line segment
Partitioning a two-dimensional rectangle
14. Write a word problem for which it makes no sense to report 1/3 • 53 as 172/3.
15. Show how to find 3/5 of 10 sevenths using discrete sets.
16. Six long distance runners get a take-out order of six pizzas for dinner. When they get home, they find that they
were shortchanged one pizza. They divide these five pizzas equally among themselves. Which of the following
expressions can be used to determine how much pizza each runner gets?
a. 6  5
b. 1/5 of 6
c. 5  6
d. 30 sixths  6
e. 1/6 of 5
B. For each of the following:
(a) Identify all described amounts A and their bases of comparison B.
(b) Write the corresponding multiplication equations of the form A = m • B.
(c) Draw and label a picture illustrating the situation.
(d) Write multiplication equations for “the other part” or “the new quantity.”
1.
2.
3.
4.
5.
6.
7.
The sales tax rate in Hamilton County, Tennessee, is 9.25%.
A shirt is on sale for 1/4 off.
Two fifths of the class are women.
The price of gas went up 10% this week.
The price of gas went down 10% last week.
Three quarters of the students at the university are undergraduates. Of these, one third are Asian.
In 1997, 23.4% of all pregnancies ended in abortion, with 55.4% of these abortions occurring within the first
eight weeks of pregnancy.
307
C. Solve the following problems.
1.
Adrian ran three fourths as far as Paula. Paula ran 24 miles. How far did Adrian run?
2.
Alison makes $60,000 more than Larry and her salary is three times his. What is their combined salary?
3. An employee gets a 10% discount on merchandise.
a. What is the discount for an item marked $79.95?
b. Determine the price the employee will pay for an item marked $147.99 by doing a single multiplication.
4. A company’s stock lost 9/10 of its value when the company went bankrupt.
a. If the stock used to be worth $20 per share, how much is it worth now?
b. If the stock is now worth $20 per share, how much was it worth before?
5.
The cost of a certain type of computer decreased by 15% this year. It used to cost two thousand dollars. How
much does it cost now?
6. The cost of gas increased by 10% this past week.
a. Last week gas cost two dollars a gallon. How much does it cost now?
b. The cost of gas is about to increase by another 20%. What will be the new cost of gas?
7.
a.
In 1999, 42.6% of accidental deaths in the United States were caused by motor vehicles. Of these, 23.7% were
people between the ages of 15 and 24. If possible, answer the following questions. If the question cannot be
answered, describe the information that would need to be known to answer the question.
What percent of accidental deaths were people between the ages of 15 and 24 who died in a motor vehicle
accident?
How many people between the ages of 15 and 24 died in a motor vehicle accident in 1999?
What percent of accidental deaths in the U.S. in 1999 were not caused by motor vehicles?
What percent of accidental deaths caused by motor vehicles were not people between the ages of 15 and 24?
What percent of accidental deaths were not people between the ages of 15 and 24 whose accidental deaths were
caused by motor vehicles?
What percent of people between the ages of 15 and 24 died in motor vehicle accidents?
b.
c.
d.
e.
f.
8. There were two thirds of a pizza left after a pizza party.
a. Suppose the tired host sat down and ate half of a pizza. How much pizza is now left?
b. Suppose the tired host sat down and ate half of what was left. How much pizza is now left?
9.
One third of the expenses for a certain business is the employee payroll. One quarter of the employee payroll is
for managers.
a. What fraction of the entire budget is for managerial employee wages?
b. What fraction of the employee budget is for non-managerial employee wages?
c. What fraction of the entire budget is for non-managerial employee wages?
10.
a.
b.
c.
d.
Seventy percent of the students at a university are women. Of the latter, 40% are 21 years old or older.
What percent of the women are less than 21 years old?
What percent of the university students are women less than 21 years old?
What percent of the students are men?
What percent of the students are at least 21 years of age?
11. In 1992, heart disease accounted for 33.10% of the 2,177,000 deaths in the U.S., while suicide accounted for
1.37% of the deaths. Of those who committed suicide, 22.67% were women.
a. Write multiplication sentences for each of the percents in this problem. State the described amounts and their
bases using English phrases, not numbers.
b. Write multiplication sentences for the “other parts” related to each percent. State the other parts and their bases
using English phrases, not numbers.
c. How many men committed suicide in the U.S. in 1992?
d. What percent of the U.S. deaths in 1992 were not due to heart disease or suicide?
____________________________________________________________________________________________
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5.4 Division in the Context of Comparisons
In this section, we investigate two more interpretations of division. Just as there are two interpretations of division
related to the basic meaning of multiplication as repeated addition, there are two interpretations of division related to
multiplication used as a means of comparison. This multiplicative relationship is summarized as follows.
Described Amount = Multiplier · Base of Comparison
If the multiplier and the base of comparison are known, we use multiplication to find the described amount. In
contrast, if the described amount is known and either the multiplier or the base of comparison is unknown, we have a
situation with an unknown factor. That is, we have a division problem.
1. Unknown Multiplier: Division as a Ratio
Described Amount =  • Base of Comparison
In the following activity, we will investigate the connection between multipliers and ratios.
Activity 5.4A
1.
Jerry ran 12 miles. Nick ran twice as far as Jerry.
a. Write the multiplicative relationship between Nick’s distance N and Jerry’s distance J.
N = ____________
b. How far did Nick run?
________________
c. What is the ratio of Nick’s distance to Jerry’s distance? Write this ratio in reduced form.
________________
2.
Mary’s salary M is three fourths of Ed’s salary E.
a. Write the multiplication sentence expressing the relationship between M and E.
M = ____________
b. If Ed’s salary is $40,000, what is Mary’s salary?
________________
c. What is the ratio of Mary’s salary to Ed’s salary? Write this ratio in reduced form.
________________
3.
Charlie bought a shirt on sale for $30. It originally cost $40.
a. State the ratio of the discount to the original price in percent form (i.e., the discount rate).
________________
b. Fill in the blank: discount = ______% of the original price
4.
A class has 8 girls and 16 boys.
a. What is the ratio of girls to boys?
________________
b. Fill in the blank using a reduced fraction: Number of girls = ____ • number of boys
5.
6.
In light of your above work, state the relationship between (a) the multiplier in
the multiplicative comparison and (b) the ratio of the amount to the base.
At Superior Tech, the tuition in 1999 was $18,500. In 2000 it was $20,000. What
was the percent increase in tuition?
________________
________________
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According to the basic definition of division as the inverse of multiplication, A = m · B implies that m = A ÷ B. The
problems in the above activity also indicate that the multiplier m is equal to the ratio of A to B. This connection
between division and ratios is the third major interpretation of division. Since the ratio of A to B is also the same
as A/B, we have the following string of equivalences.
Ratio Interpretation of Division
For B ≠ 0, the following are equivalent for computational purposes:
A ÷ B = A : B = A/B
Teaching Tip: Teachers should not assume that students will immediately recognize that the multiplier in the
multiplicative relationship between A and B is the same as the ratio of A to B. This is a surprise to many people.
Example 1:
Jerry ran 12 miles, and Nick ran twice as far as Jerry. What is the ratio of Nick’s distance to Jerry’s
distance?
Solution A
The first sentence indicates that Nick’s distance is two times Jerry’s distance. Since the multiplier in
this multiplicative relationship is 2, the ratio of Nick’s distance to Jerry’s distance is 2 to 1.
Solution B
Since Jerry ran 12 miles, Nick must have run 24 miles. The ratio of Nick’s distance to Jerry’s
distance is 24 to 12, or 2 to 1.
We have already examined a number of situations in which the ratio of two quantities is of great interest. In
situations involving multiplicative relationships, the ratio of interest is the ratio of the described amount to the base
of comparison. The ratio of A to B is often called a rate if the ratio is described as a single number. For instance,
the rate of “60 miles per hour” is the ratio of 60 miles to 1 hour. A rate is thus a ratio in which the second quantity
is expressed in terms of a single unit. A noun or adjective appearing immediately before the word “rate” is usually
a reference to the described amount. Below are some examples.
Example 2:
(a) Discount Rate = Discount/Original Price
(b) Sales Tax Rate = Sales Tax/Sticker Price
(c) Rate of Increase (or Decrease) = Increase (or Decrease)/Original Amount
If a ratio or rate is to be determined, the key is to identify the described amount and the base.
Example 3:
Peter bought a sofa on sale for $600. It originally cost $800. Find the discount rate.
The discount rate is the ratio of the discount to the original price. The discount is
$800 - $600 or $200, so the discount rate = $200/$800 = 25%.
Example 4:
Joanne paid $5.40 for an item with a sticker price of $5.00. What was the tax rate?
The tax rate is the ratio of tax to sticker price: $0.40/$5.00 = 8/100 = 8%
As the next example illustrates, we often get “two for the price of one” in situations involving ratios.
Example 5:
There are 18 girls and 6 boys in John’s class.
(a) The ratio of girls to boys is 18 to 6 or 3 : 1.
(b) The ratio of girls to the entire class: 18 to 24 = 18 ÷ 24 = 18/24 = 3/4 or 3 to 4
Mixed numbers usually need to be changed to improper fractions in order to compute simpler forms of ratios.
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Example 6:
A stock that was worth 23/4 points fell by half a point. What was the percent decrease?
Ratio of decrease to original value = 1/2 : 23/4 = 1/2 ÷ 11/4 = 1/2 • 4/11 = 4/22 ≈ 18%
“Speed” is the special name given to ratios such as distance to time or words per minute.
Example 7:
Mark drove 200 miles in 4 hours. What was his speed?
Mark’s speed = 200 mi/4 hour = 50 mi/1 hr = 50 miles per hour
2. Division as Finding the Unknown Base of Comparison
Described Amount = Multiplier • 
The fourth interpretation of division occurs when the base of comparison is unknown. These are probably the most
difficult types of division problems. It is often easier to solve such problems by setting up the multiplicative
relationship with the base of comparison as an unknown factor. The use of diagrams, the definition of multiplication,
and algebraic techniques are helpful in finding an unknown base.
Activity 5.4B
A. Solve the following problems.
1.
Peter earns three times as much money as Jim. If Peter earns $60,000, how much
money does Jim earn?
______________
2.
Maria ran one third as far as Jan. If Maria ran 6 miles, how far did Jan run?
______________
B. Connie swam two thirds as far as Jan. Let C and J represent Connie’s and Jan’s distances.
1.
State the multiplicative relationship between C and J.
______________
2.
Suppose the following line segment represents C. Underneath this line segment, draw a line segment that
accurately represents J.
3.
Suppose Connie swam 600 yards. Use your diagram to determine Jan’s distance.
4.
Explain why Jan’s distance can be found by dividing 600 yards by 2 and multiplying the result by 3.
5.
Rewrite the following as a division sentence using the basic definition of division as the inverse of
multiplication: 600 = 2/3 · 
6.
Explain how to solve the following equation by multiplying both sides of the equation by a particular fraction:
600 = 2/3 B
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C. Mandy bought a blouse at a 25% off sale.
1.
Label the parts of the diagram to the right with
“original price,” “sale price,” and “discount.”
2.
If Mandy paid $24 for the blouse, how much money did she save by buying it on sale?
Finding an unknown base of comparison is a matter of working backward from the described amount.
Example 8:
The new church hall, with an area of 4800 square feet, has three times the floor space as the old
church hall. What was the area of the old church hall?
New Church Hall
Old Church Hall
Area of new church hall = 3 · Area of old church hall
=> Area of old church hall = One of three equal parts of 4800 square feet
=> Area of old church hall = 1/3 of 4800 = 4800 square feet ÷ 3 = 1600 square feet
In other words, since the described amount is three times the base, then the base will be one third of
the described amount. Note that 1/3 is the reciprocal of 3.
Example 9:
Bobby spent two thirds of his money to rent a DVD. The rental cost $8. How much money did
Bobby have before renting the DVD?
$8
$4
$4
$4
Since $8 is two thirds of the original amount, then $8 divided by 2 must be one third of the original
amount. The original amount is three of these thirds: Original = 3 · ($8 ÷ 2) = $12.
Note that 3 · (8 ÷ 2) is the same as 8 · 3/2. Once again we have found the base by multiplying the
amount by the reciprocal of the multiplier.
As these examples illustrate, an unknown base can be reconstructed by multiplying the described amount by the
reciprocal of the multiplier.
Finding an Unknown Base
If A = c . B , then B = d . A
d
c
Algebraically, this relationship is derived as follows:
A = m·B
=>
A = m·B
m
m
=>
A = B
m
=>
B= 1 ·A
m
When the multiplier m is in fraction form, with m = c/d, then1/m is equal to d/c. So we have B = d/c · A.
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Teaching Tip: Unfortunately this division relationship between the base, the described amount, and the multiplier
is not intuitively obvious to most people. While it can be laborious to reconstruct the base using the technique
demonstrated in the above examples, students who do such reconstructions (with small numbers!) may be more
likely to solve unknown base problems correctly. Students may also be more likely to solve such problems correctly
by setting up the algebraic equation A = m · B and algebraically solving for B.
The relationship between the base and the described amount is directly connected to the fact that division is the
inverse of multiplication, as illustrated by the following diagram.
Base of Comparison
Described Amount
Multiply by m
Base
Amount
Divide by m
It is interesting that the actual process of reconstructing the base from the described amount is related more directly
to multiplying by the reciprocal of m than dividing by m. This may be one of the reasons why finding a missing base
is one of the most difficult problems in the standard school curriculum.
Example 10:
Jack owns a two-acre lot in a subdivision. It is three fourths as large as the largest lot in the
subdivision. How large is the largest lot?
Solution A
Let  represent the size of the largest lot.
2 acres = 3/4 ·  =>  = 4/3 · 2 acres = 22/3 acres
Solution B
Think this through with a diagram. Since two acres consists of three parts of the base,
we need to partition these acres into three equal parts. Do this by partitioning each acre into thirds.
Two Acres Partitioned into Three Equal Parts
One part = ⅔ acre
Largest Lot = 4 parts = 4 · (⅔ acre) = 2⅔ acres
Indirect Amounts
A complication associated with finding unknown bases is that the available information is not necessarily the amount
described by the multiplier.
Example 11:
Daisy paid $80 for a suit on sale for 20% off. How much money did she save?
Let P be the original price.
Solution A
Use the fact that 20% is equal to 1/5 to draw a diagram.
“20% off” => discount = 20% of P
=> sale price = 80% of P
=> $80 = 0.8 P
Sale Price
Discount
Original Price
=> P = $80 ÷ 0.8 = $100
=> discount = $20
Solution B
$80 is 4 fifths of the original price. Therefore 1/4 of $80, or $20, is one fifth of the original price and
also the discount.
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Example 12:
The population of Catoosa County rose by 2% in the last year. The population is now 48,400. What
was the population a year ago? Let P represent last year’s population.
2% Increase => Increase in population = 2% · P => Current population = 102% · P
=> 48,400 = 1.02P
=> P = 48,400 ÷ 1.02 ≈ 47,500
Teaching Tip: Some students have a tendency to “solve” percent problems by blindly multiplying or dividing
numbers in the problem. They hope to be lucky and stumble across the right answer. Unfortunately, luck is often in
scant supply, especially for two-step problems involving indirect amounts. In such problems it is impossible to find
the right answer by multiplying or dividing the given numbers. Teachers must help students come to understand
multiplicative relationships if students are to become competent with these very common and important problems.
5.4 Homework Problems
A. Basic Concepts
1.
a.
b.
c.
Basic Relationships
State the basic multiplicative relationship between the described amount and the base of comparison.
State the basic definition of division.
State the definition of a ratio.
2.
List the four interpretations of division discussed in this chapter.
3.
a.
c.
e.
g.
Which of the following are correct interpretations of X ÷ Y, for Y  0?
The size of a part when X is partitioned into Y equal parts
b. The number of Y’s in X
The unknown factor in the equation X = Y · 
d. The number of X’s in Y
The unknown factor in the equation Y = X · 
f. The ratio of X to Y
The unknown base for an amount X and multiplier Y
h. X/Y
4.
Which of the following are equivalent to the multiplier m in the multiplicative relationship between the
described amount A and the base of comparison B?
a. A  B
b. the ratio of A to B
c. the ratio of B to A
d. the reciprocal of A
Which of the following are equivalent to the base of comparison B in the multiplicative relationship A = 3/5 ·
B?
a. A  3/5
b. the product of A and 5/3
c. five parts, each the size of one of the three equal parts of A
5.
d. 3/5  A
e. three of five equal parts of A
f. 5/3  A
6.
a.
b.
c.
d.
e.
f.
Draw diagrams for each of the following and determine the missing numbers.
If X is four times as large as Y, then Y will be ? of X.
If X is three fourths as large as Y, then Y will be ? as large as X.
If Y increases by 20%, then the result will be ? % of Y.
If Y decreases by 20%, then the result will be ? % of Y.
If X is 50% of Y, then Y will be ? % of X.
If X is 25% of Y, then Y will be ? % of X.
7.
a.
b.
c.
d.
Consider the multiplicative comparison described by A = m • B.
Solve this equation for m.
Solve this equation for B.
m is the ratio of ? to ?
? is the base of comparison.
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8. For the multiplicative comparison, A = m • B, decide whether the following statements are true or false.
a. A is always less than B.
b. m is always a percent between 0% and 100%.
c. B must be a whole number.
d. m is the ratio of B to A.
9.
a.
b.
c.
Identify the bases and amounts for the fractions or percents in the following statements.
25% off
There will be a 10% tuition increase next year.
One fifth of the students failed the test.
10. For each of the statements in the previous problem, write a multiplication equation that includes the other part or
the new amount.
11. What is the typical base of comparison in decrease and increase problems?
12.
a.
c.
e.
Which of the following can be answered by computing 5/7 ÷ 2/5?
What is the ratio of 5/7 to 2/5?
b. How many times can 2/5 be subtracted from 5/7?
2/
5/
What is 5 of 7?
d. Find x if 5/7 • x = 2/5.
2/
5/
Find x if 5 • x = 7.
f. If 5/7 is 2/5 of another number, what is that number?
13. Invent and solve a word problem of the indicated type for each of the following.
a. 2 ÷ 1/4 (missing base)
b. 1/2 ÷ 1/4 (ratio)
1/
1/
c. 4 ÷ 2 = 8 (missing base)
d. 2 ÷ 1/2 (repeated subtraction)
B. Problem Solving
1. Seventy-five percent of the graduating seniors came to graduation.
a. Fill in the blanks: ? is 75% of ?
b. If 1200 graduating seniors were at graduation, how many did not come to graduation?
2.
A realtor sold a house for $125,000 and earned a commission of $10,000. What was her percent commission?
3.
a.
b.
c.
d.
e.
Karen bought a suit on sale for 25% off.
? is 25% of ?
? is 75% of ?
If the discount was $134.99, what was the original price of the suit?
If the original price was $134.99, what was the sale price of the suit?
If the sale price was $134.99, what was the original price of the suit?
4.
Alice saved $18.95 by using her 10% employee discount to buy a VCR. How much did she pay for the VCR?
5.
a.
b.
c.
The sales tax rate is 73/4%.
If the tax on an item is $30.42, what is the sticker price?
If the sticker price of an item is $30.42, what is the tax?
If the final price of an item is $30.42, what is the tax?
6.
Blair paid $847.99 for a sofa. The sales tax rate was 6%. What was the sticker price?
7.
At a sale, Margaret bought a blouse for $27.59 that had been originally priced at $45.99. What was the discount
rate?
8. Mary makes 3/4 as much money as John. John’s salary is $46,000.
a. What is the ratio of Mary’s salary to John’s salary?
b. What is Mary’s salary?
9.
Seth had to pay a 10% penalty when he made a late payment. The penalty was $15. How much was the final
bill?
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10. Ben invested three fourths of an inheritance. He bought a boat with the remaining money. If the boat cost
$6000, how much money did he inherit?
11. Rachel has 18 feet of string and cuts it into half-foot lengths for a project.
a. How many pieces of string does she now have?
b. State the division sentence that yields the answer to this question.
12. April has 18 feet of string and cuts it in half for a project.
a. How many pieces of string does she now have, and how long are they?
b. State the division sentence that yields the answer to this question.
13. A half acre of land is sectioned off into 40 garden plots of equal size. How big is each plot?
14. A square mile is equal to 640 acres. How many square feet are in an acre?
[Hint: A square mile is 5280 feet by 5280 feet.]
15.
a.
b.
c.
One third of the crew of a ship got seasick during a storm.
If there were 6 crewmen, how many got sick?
If there were 6 sick crewmen, how many crewmen did not get sick?
If there were 6 crewmen who did not get sick, how many crewmen were there altogether?
16.
a.
b.
c.
Twenty percent of a class made A’s.
If 40 students made A’s, how many students did not make A’s?
If 40 students did not make A’s, how many students were in the class?
If there were 40 students in the class, how many did not make A’s?
17. A stock lost one tenth of its value in 2000 and one quarter of its remaining value in 2001. What was the stock
worth after these changes, relative to its value at the beginning of 2000?
18.
a.
b.
c.
The price of a computer dropped 10% in 1998 and another 15% in 1999.
If the computer cost $2449 in 1997, how much did it cost in 1999?
If the decrease in price was about $150 in 1998, what was the decrease in price in 1999?
What was the overall percent change in the price of computers in these two years?
[Percent change is the ratio of the change in price to the original price.]
19. Berta paid $31.47 for a pair of pants on sale for 30% off. How much money did she save by buying the pants on
sale?
20. Hakeem paid $1407.24 for a bedroom suite, including an 8.25% sales tax. How much sales tax did he pay?
21. After a 7% increase, full-time tuition is now $1349. What was the old tuition?
22. In 1991, the United States consumed about ten times as much energy as India, even though India has more than
three times as many people as the United States. The U.S. consumed about 80 quadrillion Btu. (“Btu” is an
abbreviation for British thermal unit, a measure of energy.)
a. How much energy did India consume?
b. How much energy did an average American consume, compared to an average Indian?
23. Sarah inherited two thirds of her mother’s estate. She decided to give one tenth of her inheritance to charity. If
she gave $1500 to charity, how much money did she inherit?
24. A teacher sent 15 students to the library. This was three fourths of her class. How many students are still in the
classroom?
25. John inherits 5/7 of his mother’s estate. He invests 2/5 of his inheritance and spends the rest on a trip to Alaska.
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a. What fraction of the entire estate did he invest?
b. What fraction of his inheritance did he spend on his trip to Alaska?
26. John is in charge of 5/7 of his mother’s estate. He invests 2/5 of the entire estate in Company X and the rest of
the estate for which he is responsible in mutual funds. What fraction of his mother’s estate are in mutual funds?
27. John inherits 2/5 of a small parcel of land. His inheritance amounts to 2/7 of an acre. What is the total acreage of
the small parcel of land?
28. In 1992, the world record for the 1500 meter run was 3 min 40.12 sec. The world record for the 1500 meter
freestyle swim was 14 min 43.48 sec. How much faster is the world record in running compared to the world
record in swimming?
a. Estimate answers using (1) subtraction and (2) division.
b. Find exact answers using (1) subtraction and (2) division.
____________________________________________________________________________________________
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____________________________________________________________________________________________
5.5 Proportional Reasoning
In this section, we explore constant ratios in greater depth.
Activity 5.5A
A. An ad in the produce section of the supermarket reads, “Two watermelons for $3.00.”
1.
Answer the following questions, supporting your answers with appropriate diagrams.
a. How much will six watermelons cost? ________
2.
b. How much will five watermelons cost?
_________
Let: C = cost of watermelons and W = number of watermelons.
a. Complete the following table.
W
0
1
2
d. Graph your ordered pairs.
5
6
10
C
b. Express the relationship between C and W using
multiplication.
c. Express the relationship between C and W using
ratios.
e. Find the slope of the line defined by these points.
B. Answer the following. Assume this is a one centimeter grid.
1.
2.
Identify two sets of rectangles with the same shapes.
Set 1: ______________
Set 2: ______________
A
B
C
Complete the following tables for each set, including the
ratios of corresponding sides of rectangles in each set.
Use fraction form for your ratios.
Set 1
Set 2
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Rectangle
Short Side
Long Side
_______
_______
________
________
________
________
Ratio
________
________
Rectangle
Short Side
Long Side
_______
_______
________
________
________
________
Ratio
________
________
D
F
E
G
H
3.
Look for a pattern and make a generalization about the ratios of the corresponding sides of “look alike”
rectangles.
4.
For each of the above sets of two rectangles, find the ratio of the larger area to the
smaller area. Use fraction form. [Hint: These ratios are not what you might expect.]
Set 1: _________
Set 2: _________
5.
Fill in the following table. Assume the smaller cube is 1 cm by 1 cm by 1 cm
and the larger cube is 2 cm by 2 cm by 2 cm. Include units.
Smaller Cube
Larger Cube
Ratio
Length of a side
Area of a face
Volume of cube
Proportional Relationships and Their Connection with Multiplicative Relationships
If the ratio of two related variable quantities A and B remains constant even as the two quantities change, then A and
B are said to be proportional. For example, the ratio of the cost to the number of watermelons at a supermarket
probably remains constant even as the cost and number change, the ratio of sales tax to sticker price remains constant
for different prices, and the ratio of the velocity of a free falling object to the time it has been falling is a constant.
Example 1:
If a pound of asparagus costs $3.00, then 2 pounds will cost $6.00, half a pound will cost $1.50, a
third of a pound will cost $1.00, and so on. The constant in these situations is the ratio of weight to
cost: $3.00/1 lb = $6.00/2 lb = $1.50/0.5 lb = $1.00/(⅓ lb). All of these are ratios of 3 to 1.
The equation Y/X = A/B is equivalent to the equation Y = A/B • X. Thus two quantities are proportional if and only if
one quantity is a constant multiple of the other. This multiplicative relationship is exactly the type of relationship we
studied in previous sections. In other words, quantities with a multiplicative relationship also have a proportional
relationship, and vice versa. As we shall see, some problems are easier to solve using a proportion while others are
easier to solve using the multiplicative relationship.
Example 2:
Reconsider the asparagus that cost $3.00 a pound. Let W represent the weight of the asparagus you
are buying and C represent the cost of this asparagus. Presumably the ratio of price to weight is
constant, so we have $3.00/1 pound = C/W. The equivalent multiplicative relationship is C = 3 · W.
Proportionality and Similar Figures
Proportional relationships are common in geometry. Similar figures were defined earlier as figures that have the
same shape but not necessarily the same size. Now we can state more precisely that similar figures are such that
their corresponding sides are proportional and their corresponding angles are congruent.
3
Example 3:
The following two right rectangular solids are similar.
This means that the ratio of the corresponding heights
of these solids is the same as the ratios of the corresponding lengths and the corresponding widths.
6 7.5 3
=
=
4
5
2
2
6
4
7.5
4
5
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Example 4:
The ratio of the circumference to the diameter of a circle
is constant, regardless of the size of the circle.
C/D
=
d
c/d
D
This ratio is the irrational number π.
C/D = π
=> C = πD.
c
C
Proportionality in One, Two, and Three Dimensions
Areas and volumes of similar shapes have predictable relationships.
2nd
Example 5:
In a little league baseball diamond, it is 60 feet from home plate to
first base. In the major leagues, this distance is 90 feet. Find the
ratio of these distances and the ratio of the areas of these infields.
(The infield is the square area bounded by the baselines.)
3rd
1st
Ratio of distances = 90 ft/60 ft = 3/2 = 1.5
2 2
Ratio of infield areas = 90 ft /602 ft2 = 8100/3600 = 9/4 = 2.25
home plate
Thus a major league base runner has to run one and a half times as far as a little leaguer to get to first
base; a major league infielder also has to cover over twice as much area as a little leaguer.
Example 6:
A small nougat of chocolate candy measures 1 cm by 1 cm by 3 cm and weighs about half an ounce.
A larger nougat has dimensions that are double the dimensions of the smaller nougat. How much
does the larger nougat weigh?
As the diagram illustrates, the larger nougat has
a volume that is 8 times the volume of the smaller
nougat, so it weighs 8 times as much as the ½ oz.
nougat, or about four ounces.
1 cm by 1 cm by 3 cm
2 cm by 2 cm by 6 cm
These examples illustrate the following relationships among ratios in one, two, and three dimensions.
Dimension
1-dimensional
2-dimensional
3-dimensional
Example 7:
Type
Length
Area
Volume, Weight
Ratio
k:1
k2 : 1
k3 : 1
Example
3:1
9:1
27 : 1
Suppose a 5-foot tall woman weighs 100 pounds. How much would a 6-foot tall woman with the
same shape as the shorter woman weigh?
The ratio of one-dimensional heights is 6 to 5 or 6/5. Since weight is associated with volume, the
3
corresponding ratio of three-dimensional volumes will be 6 /53 or about 1.73 to 1. Thus the weight of
the taller woman with the same shape is about 1.73 · 100 pounds or 173 pounds.
Teaching Tip: Most students are amazed by the above relationships among length, area, and volume. Apparently
our intuitions are working against us here. Thus students should be given lots of experiences comparing one-, two-,
and three-dimensional characteristics of similar figures and shapes. It is a good idea to use manipulatives such as
grid paper and building blocks for this purpose.
Within and Between Ratios
Situations involving constant ratios involve four quantities. There are two major ways to arrange these quantities.
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Example 8:
The cost of 16 ounces of tomatoes is $1.79. If the ratio of cost to weight is constant, what is the cost
of 12 ounces of tomatoes? Let C represent the cost of 12 ounces of tomatoes.
a. Use the ratios of cost to weight: $1.79/16 ounces = C/12 ounces
b.
Use the ratios of corresponding quantities: $1.79/C = 16 ounces/12 ounces
A ratio of two quantities within the same situation is a within ratio. For example, the above ratios of cost to weight
are within ratios. The ratio of weight to cost is also a within ratio. A ratio of corresponding quantities in different
situations is a between ratio. In the above example, the ratio of the first cost to the second cost is a between ratio;
so is the ratio of the first weight to the second weight.
Solving Proportions
An equation of the form A/B = C/D, in which two ratios are set equal to each other, is called a proportion. In
situations involving constant ratios, we often know three of the four numbers in a proportion and are interested in
figuring out the fourth. Below are three common ways of doing so.
1. The Unit Rate Method
Example 9:
A 15-oz can of clams costs $3.00. If the unit price is constant, how much should a 22-oz can cost?
The unit price is the cost per ounce. For the first can of clams, the unit price is
$3.00 ÷ 15 oz = 20¢ per ounce. So 22 oz • 20¢ per oz = $4.40.
In general, the unit rate for two proportional quantities is the amount of the first quantity A per one unit of the
second quantity B. It is simply the reduced ratio of A to B found by calculating A ÷ B. This corresponds to the
multiplier m in the multiplicative relationship A = m • B.
Teaching Tip: Send your students off to supermarkets that list unit prices to compare the unit prices of items
packaged in varying sizes (e.g. cans of clams).
Constant ratios are the basis for creating and using scale models such as maps and model airplanes. The unit rate
method of determining corresponding values is particularly useful in these situations because multiple values often
need to be calculated.
Example 10:
On a backpacker’s map, every two inches represents five miles. On the map, the distances from the
start to the end of two trails are 7'' and 41/2''. How long is each trail?
If two inches represents five miles, then one inch represents 21/2 miles.
Length of first trail = 7 inches • 21/2 miles per inch = 171/2 miles
Length of second trail = 41/2 inches • 21/2 miles per inch ≈ 11miles
2. The Scale Factor/ Factor of Change/ Divisor of Change Method
Example 11:
Cantaloupes are three for five dollars. How much will six cantaloupes cost?
2
3 cantaloupes = 6 cantaloupes
$5
2 • $5
=>
Six cantaloupes will cost $10.
2
Solving the cantaloupe problem is a matter of observing that 3/5 is the same as 6/10. This is an application of the
Fundamental Property of Fractions: A/B = nA/nB for any nonzero number n. The number n is referred to as the scale
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factor or factor of change. Since the FPF also states that A/B = A÷n/B÷n, proportions can also be solved using a
divisor of change. This method is very handy if the factor or divisor of change is a small whole number.
As the following example shows, sometimes two factors of change can be used to find an answer.
Example 12:
Right triangles A and B are similar. What is x?
x
10''
10 = 5 and 5 = x
6
3
3
9
=>
x = 15 inches
6''
9''
A
B
The scale factor method works well only if the numbers are compatible, that is, when one number is a whole number
multiple of another, such as 9 and 3.
Teaching Tip: Students become familiar with scale factors when they are learning to add fractions with different
denominators—although they usually do not know the process by this name. For instance, they find the numerator
in 5/4 = ?/12 by identifying the scale factor as three. Later on, teachers rewrite “ 5/4 = ?/12” as “5/4 = x/12” and call it
a proportion. Rather than insisting that students solve this equation using some other technique, teachers should
build on what students already know and encourage them to apply the scale factor method when appropriate.
3. The Cross Products Algorithm
A third way to solve proportions is to use the following theorem and a little algebra.
Cross Products Theorem: The equation A/B = C/D is equivalent to AD = BC, for B  0 and D  0.
Proof:
A = C
B
D
=>
BD . A = BD . C
1
B
1
D
=>
BD . A = BD . C
1
B
1 D
=>
AD = BC
AD and BC are called cross products. Sometimes the process of converting A/B = C/D to the equivalent equation
AD = BC is called “cross-multiplying.”
Teaching Tip: Unfortunately the phrase “cross-multiplying” is also used to describe the shortcut for dividing a
fraction by a fraction. W hen different processes are referred to by the same name, students often confuse the
results. With a proportion, the result of “cross-multiplying” is another equation; with division of fractions, the
result is another fraction: 2/3  7/x is equal to 2x/21 but 2/3 = 7/x is equivalent to the equation 2x = 21. To avoid
confusion, it is better not to refer to the division of fractions as “cross-multiplying.”
As the next example illustrates, this algebraic approach to solving proportions is useful when dealing with more
difficult numbers.
Example 13:
A nurse knows that the dosage of a certain antibiotic is 30 ml for an 80-pound child. What should
the dosage be for a 105-pound child?
80 lb. needs 30 ml
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105 lb. needs x ml
Solution A
Using cross products
(1) Within Ratios Proportion
(2)
30 ml = x ml
80 lb
105 lb
Between Ratios Proportion
x ml = 105 lb
30 ml
80 lb
Both of these proportions lead to the following equation.
80x = 30 • 105
Solution B
x = 30 ml • 105 lb ≈ 39 ml
80 lb
Using standard equation solving techniques (and one less step than cross-multiplying)
x ml = 30 ml
105 lb 80 lb
Solution C
=>
=>
x = 105 • 30 ≈ 39 ml
80
Using unit rates
30 ml/80 lb
= 0.375 ml per pound
=>
105 pounds . 0.375 ml ≈ 39 ml
pound
As the above example illustrates, a variety of methods can be used to find an unknown in a proportional relationship.
The main challenge is setting up the ratios correctly. This is greatly facilitated by paying attention to units and using
within ratios. For instance, if the ratio on one side is milliliters to pounds, then the ratio on the other side must also
be milliliters to pounds. Between ratios can also be used, but care must be taken so that the quantities in the two
numerators (and the two denominators) come from the same situation. Another way to guarantee correct results is to
set up operations so that units divide out correctly. We will explore unit cancellations in the next section.
Teaching Tip: Proportional reasoning is far more than the ability to follow procedures for solving proportions. It
is important to develop students’ conceptual understanding of proportional relationships in a wide variety of
settings rather than simply focusing on procedures for solving proportions.
Proportions and Multiplicative Relationships
As we have already discussed, quantities that are proportional also have a multiplicative relationship. This means
that problems can often be solved two ways, either with a proportion or a multiplication sentence.
Example 14:
Matt paid only $240 for a refrigerator at a 40% off sale. What was the original price P?
discount = 40
“40% off” =>
original price 100
=>
=>
sale price = 60
P
100
$240 = 60
P
100
=>
P = $240 · 100 = $400
60
Teaching Tip: While some problems involving percents lend themselves to solutions using proportions, not all
problems do so. It is important for students to know how to describe proportional relationships both
multiplicatively and with ratios.
Activity 5.5B
1.
A school had a 20% increase in enrollment and now has 425 students. How many more students are enrolled at
the school now than before? Solve this problem two ways:
a. Using a proportion
b. Using a multiplication sentence
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2.
The photocopying machine is set so that the new dimensions will be 300% of the original dimensions. The
original figure is a 2'' by 3'' rectangle.
a. Find the dimensions of the enlarged image.
_____________________
b. Find the ratio of the area of the enlarged image to the area of the original figure.
3.
A woman who is five feet tall weights 100 pounds. Another woman who has the same general build is 5’6” tall.
About how much does the second woman weigh? [Hint: Weight is related to volume.]
4.
A ranger wants to estimate the number of fish in a small lake. Her first step is to catch and tag 20 fish. Then she
returns these fish to the lake. Later she catches 40 fish. She finds that five of these fish are tagged. If she
assumes that the proportion of tagged fish in the lake’s fish population is about the same as in her second catch,
about how many fish are in the lake?
5.5 Homework Problems
A. Basic Concepts
1.
a.
b.
c.
d.
e.
f.
g.
Suppose A and B are proportional quantities. Which of the following must be true statements?
A and B remain constant.
The ratio of A to B remains constant for corresponding values of A and B.
A is a constant multiple of B.
B is a constant multiple of A.
A and B have a multiplicative relationship.
If A increases by 2 units, so will B.
If A doubles, so will B.
2. Similarity
a. Similar figures have the same ? but not necessarily the same ?
b. Two figures are similar if their ? sides are ?
3. An 18-ounce can of tomatoes costs $1.89.
a. If the price per ounce is constant, how much will a 12-ounce can of tomatoes cost? Solve this problem using a
proportion containing within ratios.
b. Find and use the unit price (cost per can) to find the cost of the 12-ounce can.
4. Ears of corn are advertised as “10 for $2.”
a. Find and use the unit price to determine the cost of 8 ears of corn.
b. Use the Scale Factor/Divisor Method to find the cost of 15 ears of corn.
5.
Solve the following using the Scale Factor Method.
a. 4 = 12
5
x
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b. 6 = x
9
3
c. 48 = 24
150 x
d. 15 = 10
6
x
6.
Explain how the Scale Factor Method of solving proportions is based on the Fundamental Property of Fractions.
Include an example.
7.
Informally stated, the basic principle of equation solving is as follows: “Doing the same thing to both sides of
an equation produces an equation with the same solutions as the original equation.” For instance, if 3 is
subtracted from both sides of x + 3 = 5, the resulting equation will have the same solution as the original
equation. What must be done to both sides of the proportion A/B = C/D to produce the equivalent equation
AD = BC?
8.
Suppose a child is having a hard time grasping the idea of constant ratios. She thinks that the ratio of 8 to 5 is
the same as the ratio of 9 to 6 and that the ratio of 4 to 1 is the same as the ratio of 6 to 3.
a. What is this child’s misperception?
b. Use the basic definition of a ratio and pictures to help this child see that 4 : 1 is not the same as 6 : 3.
9a. If Y = 3 · X, what is the ratio of Y to X?
b. If P and Q are proportional quantities with P/Q equal to 4/3, what is the value of the multiplier in the equivalent
multiplicative relationship, P = m · Q?
10a. State the definition of π.
b. Using a measuring tape or a ruler and string, measure to the nearest millimeter the diameter and circumference
of a handy large circular item (a wastebasket, the rim of a bowl, a flower pot, etc.) Then find the ratio of the
diameter to the circumference.
c. Find the difference between your ratio and π to three decimal places.
d. Find your percent error, the ratio of the error (from part c) to the actual value.
11. Suppose the lengths of all sides of a square are tripled. Use a labeled and carefully drawn illustration to show
the effect on the area of the square. It may be helpful to use grid paper.
12. Higher Dimensional Relationships
a. Carefully draw representations of two cubes, one with an edge of length 1 cm and the other with an edge of
length 3 cm.
b. What is the ratio of the areas of the front faces of these cubes?
c. What is the ratio of the volumes of these cubes?
13. If the ratios of the edges of two cubes is p : q, state the following.
a. Ratio of the areas of the faces of these cubes
b. Ratio of the volumes of these cubes
B. Suppose cans of beans are advertised at “5 for $4.” Assume the ratio of cans to cost remains constant. Let N
represent the number of cans and C the cost of N cans.
1. Find the unit rate.
2. Make a table of six pairs of values for N and C.
3. Graph your ordered pairs on graph paper.
4. Find the slope of the line formed by your graph and compare it to the unit rate. Explain any similarities.
5. State the relationship between N and C in two ways:
a. Using ratios
b. Using multiplication
C. Problem Solving
1.
a.
b.
c.
Avocados are advertised as “4 for $3.” Find the cost of six avocados in three ways:
Unit rate method
Factor/ divisor of change method
Setting up a proportion and cross-multiplying
2.
The prescribed dosage of a certain antibiotic is 30 ml for a 50-pound child. Answer the following questions
using the method stated in parentheses.
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a. How much antibiotic should be given to a 75-pound child? (divisor/ factor of change)
b. How much antibiotic should be given to an 87-pound child? (within ratios proportion)
c. How much antibiotic should be given to an 113-pound child? (between ratios proportion)
3.
a.
b.
c.
d.
e.
The two rectangles to the right are similar.
Construct a proportion using within ratios.
Construct a proportion using between ratios.
Use cross products to find x.
Use the factor of change method to find x.
Find the ratio of the areas of these rectangles.
4.
Grocery store #1 advertises 15-oz. cans of pork and beans at “4 for $1.” Grocery store #2 advertises a 28-ounce
can of pork and beans for 59¢. Determine the better deal using (a) unit rates, and (b) a factor of change.
5.
On a map, two inches represent 9 miles. If two points are 3.5 inches apart on the map, how far apart are they in
actuality?
6.
Grocery store #1 advertises “1/2 gallon Gatorade: 3 for $5.” Grocery store #2 advertises
“64-ounce Gatorade: 2/$3.” Determine the better deal by using (a) unit rates, and (b) a factor of change.
8'
24'
x
45'
7. Justify your answer for the following using labeled diagrams.
a. One right triangle has legs of length 9'' and 12''. Another right triangle has legs of length 6 cm and 8 cm. Are
these triangles proportional?
b. One triangle has sides of length 9'' and 12''. Another triangle has sides of length 6 cm and 8 cm. Are these
triangles proportional?
8.
a.
b.
c.
Two boxes are similar. The shortest side of the larger box is three times the shortest side of the smaller box.
What is the ratio of the longest side of the larger box to the longest side of the smaller box?
What is the ratio of the bases of the two boxes?
What is the ratio of the volumes of the two boxes?
The pitch of a roof is a measure of the roof’s steepness. It is the ratio
of the length of the vertical to the horizontal leg in the right triangle
N
formed underneath the roof. Construction workers describe the pitch
of a roof in the form “N and 12,” which means the ratio of N to 12.
12
Draw diagrams on grid paper for each of the following problems.
a. Draw a roof with a pitch of 8 and 12.
b. The pitch of a roof is to be 5 and 12. If the vertical beam is to be 8 feet, how long should the horizontal beam of
the truss be? (The truss is the roof support represented by the isosceles triangle in the above diagram.)
c. A rectangular house is to be built 40 feet wide and 60 feet long. Find the dimensions of the trusses needed
for this house, if the pitch is to be 5 and 12.
9.
10. The grade of a road refers to the ratio V/H of the vertical to the
horizontal change from one point on the road to another. It is
often expressed in percent form because it is usually a small
fraction.
V
H
a. As I-24 comes off the Cumberland Plateau in southeastern Tennessee, there are large signs warning truckers of
an upcoming 7% grade. Explain the meaning of this number.
b. If one leg of a right triangle is very small compared to the other, then the hypotenuse of the triangle has almost
the same length as the longer leg. Use this fact to estimate the height (in feet) of the Cumberland Plateau above
the valley, if it takes about three miles to drive down the 7% grade to the bottom of the mountain.
_______________________________________________________________________
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