My Math Path: 4, 5 and 6 – Decimal numbers, Common fractions
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MY MATH PATH 4 – DECIMAL NUMBERS
Contents
Learning outcomes .....................................................................2
Glossary .....................................................................................3
Introduction ................................................................................4
Section 1: Place value and rounding numbers ........................10
Exercise 1 .......................................................................12
Section 2: Comparing numbers ...............................................14
Exercise 2 .......................................................................16
Section 3: Rounding numbers .................................................17
Exercise 3 .......................................................................20
Section 4: Operations with decimals.......................................22
Exercise 4 .......................................................................23
Section 5: Multiplying whole numbers and decimal
fractions ..........................................................................26
Exercise 5 .......................................................................27
Section 6: Dividing whole numbers and decimal
fractions ..........................................................................29
Exercise 6 .......................................................................32
Section 7: Order of operations ................................................35
Exercise 7 .......................................................................37
Practice Test .............................................................................38
Answers....................................................................................40
MY MATH PATH 4
Decimal Fractions
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Learning outcomes
Being able to work with whole numbers, decimals and fractions is important in everyday life.
Estimating an answer to a problem helps put the problem in context. This process should be
part of any problem solving activity. Checking answers and solutions to problems is equally
important.
When you have completed this Unit, you should be able to:
round numbers
calculate and solve problems involving adding, subtracting, multiplying and dividing
whole numbers, and decimals fractions
check that answers and solutions to problems are reasonable in the context of the given
questions
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Decimal Fractions
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Glossary
Difference
The answer to a subtraction question is called the difference.
Equivalent
Equivalent quantities are equal.
Factor
A number that divides evenly into another number. E.g., 2 and 3 are factors of 6.
Prime number
A prime number is divisible only by itself and one. The number one is not a prime number.
The first few prime numbers are 2, 3, 5, 7, 11, ...
Product
The answer to a multiplication question is called the product.
Quotient
The answer to a division question is called the quotient.
Sum
The answer to an addition question is called the sum.
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Decimal Fractions
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Introduction to Fractions
This is the beginning of an adventure with numbers that represent part of the whole thing. These
numbers are fractions.
The idea of fractions may be very comfortable to most of us because our minds are used to dealing
with parts in our everyday life. Look at the pictures and use a fraction to answer the questions.
one quarter
1
4
one third
1
3
one half
1
2
two thirds
2
3
three quarters
3
4
How much gas is left?
a)
__________ of a tank
MY MATH PATH 4
b)
Decimal Fractions
__________ of a tank
4
This full bubble gum machine is the
whole thing.
Do we need to fill up the bubble gum machine?
c)
No, it’s ___________ full.
MY MATH PATH 4
d)
Yes, it’s ___________full
Decimal Fractions
5
Common Fractions
Common Fractions are written with two numbers, one above the other, with a line in
between. The line may be straight – or on an angle /
2
3
2/3
(it also means 23)
The denominator is the bottom number. The denominator tells how many equal parts
there are in the whole thing.
2
3
numerator
denominator
The numerator is the top part in a common fraction. The numerator tells how many of
the equal parts we are actually describing or talking about.
This pizza has been cut into eight pieces,
all the same size (equal). The denominator
to use while talking about this pizza is 8.
The 4 numerator will be the exact number
of pieces of the pizza you are describing.
This is 1 pizza, and that is the whole
8
thing. If someone ate all 8 pieces or
8
(eight-eights) that person ate 1 pizza. 3 of
3
the pieces would be of the pizza.
8
You will learn more about common fractions in unit 3.
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Decimal Fractions
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Decimal Fractions
What is a decimal fraction?
Decimal Fractions are another way of considering parts of the whole thing – and the
whole thing is one. You use decimal fractions everytime you think about money! The
dollars are written as whole numbers; the cents are written as a decimal fraction of a dollar.
A decimal fraction does not have a denominator written underneath it. Instead a decimal
point (.) separates the whole number from the fraction. Our number system is called a
decimal system because it is based on the number ten (“deci” is the Latin word for ten). So in
decimal fractions the whole thing is divided into tenths; the tenths are divided by ten to make
hundredths; the hundredths are divided by ten to make thousandths and so on.
Decimal fractions are uses a great deal in our everyday lives, especially in money and
measurement.
$12.24
3.5 kilometres to the store
2.6 metres of material
1.8 kilogram of pasta
As you know, fractions describe part of the whole thing – a fraction is smaller than 1. And as
you know, 1 (the whole thing) can be many things. For example, it can be
one dollar
one city
one school
one paycheck
one year
one second
one pregnancy...
So a decimal fraction may represent part of a pregnancy, part of a second or part of anything
you want.
Here are the characteristics of a decimal fraction:
A decimal point . separates whole numbers from fractions. A decimal fraction starts with
a decimal point.
.1
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.34
0.5
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In a decimal fraction, the denominator is not written. Remember that the denominator in
a common fraction is the bottom number and tells how many equal parts there are in the
whole thing.
2
3
denominator
But in a decimal fraction the denominator is understood. We tell the size of the denominator
by looking at how many numerals are placed after the decimal point.
Decimal denominators are always ten or ten multiplied by tens.
0.4
0.44
0.444
0.4444
0.44444
0.444444
has a denominator of 10
has a denominator of 100
has a denominator of 1 000
has a denominator of 10 000
has a denominator of 100 000
has a denominator of 1 000 000
A whole number and a decimal fraction can be written together. This is called a mixed
decimal fraction.
4.35
100.47
$12.39
Every whole number has a decimal point after it, even though we usually do not bother to
write the decimal point unless a decimal fraction follows the number. We can also put
zeros to the right of the decimal point of any whole number without changing its value.
3=
3. = 3.000000
275 = 275. = 275.0
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Decimal Fractions
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Reading and Writing Decimal Fractions
Common fractions with a denominator of 10 are written as a decimal fraction with one place
to the right of the decimal point. This is the tenth place.
2
= .2 = two tenths
10
6
= .6 = six tenths
10
Introduction Exercise One Write each fraction as a decimal fraction and in words.
a.
4
=
10
b.
1
=
10
=
c.
9
=
10
=
.4
=
four tenths
When we write decimal fractions, a zero is usually placed to the left of the decimal point to
show there is no whole number. This zero keeps the decimal point from being “lost” or not
noticed.
Introduction Exercise Two Write each decimal fraction as a common fraction and
in words.
a.
0.3
=
b.
0.5
=
c.
0.7
=
3
10
three tenths
Solutions are on p.37
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Decimal Fractions
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Section 1: Place value
Place value
Words are formed with letters. Numbers are formed with digits. The ten digits of our
number system are:
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
By stringing these digits together we can form numbers. The position that a digit occupies in
a number tells us how many ones, tens, hundreds, thousands (and so on) that the number
contains. If the number has a decimal point, then the position of the digits after the decimal
point indicates how many tenths, hundredths, thousandths (and so on) the number contains.
The place values or position values in a number are shown in Figure 1.
Figure 1 Place values
Example 1
Write the place value for each underlined digit.
a. 43 209
b. 6.053 21
c. 635 096.58
b. thousandths
c. hundred thousands
Solution
a. tens
Example 2
Write 23 805.90 in expanded form.
Solution
23 805.90 = 20 000 + 3 000 + 800 + 5 +
9
0
+
10 100
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Example 3
Write three million in standard notation.
Solution
To write a number in standard notation means to write it using only digits. Three million in
standard notation is 3 000 000.
Now complete Exercise 1 and check your answers.
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Exercise 1
1.
Write the place value for each underlined digit.
a. 852
_______________________
b. 43.921
_______________________
c. 0.056
_______________________
d. 163 940
_______________________
e. 0.000 502 _______________________
2.
3.
4.
Underline the digit for the place value named.
a. tens
842.56
b. tenths
842.56
c. hundredths
68 923.05
d. millions
843 907 000
e. ten thousandths
63 521.009 842
Write in expanded form.
a. 823
_______________________________________
b. 8.23
_______________________________________
c. 9 605
_______________________________________
d. 0.087
_______________________________________
Luna had 6 one hundred dollar bills, 3 tens and 8 loonies. How much money did she
have?
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5.
Bill had 5 ten dollar bills, 3 dimes and 2 pennies. How much money did Bill have?
6.
Ida found 4 pennies and 1 loonie. Together with her ten dollar bill, how much money
does Ida have?
7.
Dan wants $203.41. How many hundred dollar bills, ten dollar bills, loonies, dimes
and pennies should he receive?
8.
A certain company lost 18.3 million dollars last year. Write this number in standard
notation.
Answers are on page 40.
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Section 2: Comparing numbers
There are only three possibilities when comparing numbers. Either one number is equal to
(=) another number, greater than (>) another number or less than (<) another number.
When comparing decimal fractions, it is often a good idea to write both numbers with the
same number of digits after the decimal point. This is possible because placing zeros after
the last decimal digit will not change the value of the number. For example:
63.5 = 63.50
63.5 = 63.500
63.5 = 63.5000
and so on.
We can do the same with whole numbers, but only after placing a decimal point at the end of
the whole number. For example:
804 = 804
804 = 804.0
804 = 804.00
804 = 804.000
and so on.
Example 1
Use >, < or = to compare each pair of numbers.
a. 0.002
0.01
b. 69
69.00
c. 305
30.5
Solution
a. Since 0.002 has three decimal digits, write 0.01 with three decimal digits, by putting
another 0 after the 1. Comparing:
0.002 < 0.010
b. 69 and 69.00 have the same value. They are said to be equivalent:
69 = 69.00
c. 305 is greater than 30.5. Notice that a decimal could be placed at the end of 305.
305 > 30.5
Notice how greater than or less than signs always point to the smaller number,
such as 13 > 5 or 5 < 13.
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Example 2
Find the smallest of the three numbers below:
5.0
5.01
4.999
Solution
The equivalent numbers with three decimal digits are:
5.000 5.010 4.999
The smallest number is 4.999.
Now complete Exercise 2 and check your answers.
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Exercise 2
1.
Use >, < or = to compare the numbers.
a. 9 919
9 991
b. 0.5
0.50
c. 6.2
6.02
d. 84.009
84.1
e. 31 280
2.
31 280.0
Find the smallest number in each set of numbers.
a. 16
1.6
16.0
b. 0.040 20
c. 8.70
e. 199.0
3.
4.
0.040 18
8.700 000
d. 13 562
160
13 625
99.999
0.041
8.07
13 265
8.7
16 235
0.999 999 999
Use an “is equal to” (=) or an “is not equal to” () symbol to compare the numbers.
a. 0.02
0.20
d. 842
8420.0
b. 6.3
63
e. 107
1.07
c. 9.00
9
f. 98 761
98.761
Jill bought gas for 49.9 cents per litre. Jack said he bought gas for $0.499 per litre.
Who paid the most for gas?
Answers are on page 40.
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Section 3: Rounding numbers
Estimation can be used to calculate the approximate cost of something.
Example
You and a friend are having dinner at a restaurant. As you look at the menu, you try to
estimate the cost of your meal.
Main Course
Roast Beef
Chicken
Spaghetti
Vegetarian Lasagna
$13.95
$11.50
$ 9.75
$10.99
Desserts
Cheesecake
Apple Pie
Ice Cream
$ 4.25
$ 3.90
$ 2.75
Beverages
Milk
Coffee
Tea
Soft Drinks
$
$
$
$
1.50
0.99
0.99
1.55
You choose chicken, apple pie and coffee. Your friend chooses lasagna, ice cream and a soft
drink.
To estimate the cost of each person’s meal, you can round each price up or down. In this
example, if you round up, your estimate will be higher than the actual cost.
Your meal
Chicken
Apple pie
Coffee
Total estimated cost
Your friend
$12
$ 4
$ 1
Lasagna
Ice cream
Soft drink
$17
$11
$ 3
$ 2
$16
Note that estimates are not exact answers and may vary depending on whether you round up
or down.
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Decimal Fractions
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No one would ever report his or her age as being 30.859 years old. Knowing that the person
is approximately 30 years old is usually sufficient. Most values like age can never be known
exactly. To make things simple, we often round numbers to a convenient place value.
Example 1
The population of Canada is 28 463 900 people. Round to the nearest million.
Solution
Step 1: Underline the millions digit
28 463 900
Step 2: If the next digit after the underlined digit is a 0, 1, 2, 3, or 4, then do not change the
underlined digit. If the next digit is a 5, 6, 7, 8, or 9, then add 1 to the underlined
digit. Since the next digit is a 4, the number is still 28 463 900.
Step 3: Change all the digits after the underlined digit to zeros. Then remove the underline.
The population of Canada is 28 000 000.
The steps required to round any decimal fraction are as follows:
Step 1: Underline the digit to which the number is being rounded.
Step 2: If the next digit after the underlined digit is a 0, 1, 2, 3, or 4, then do not change the
underlined digit. If the next digit is a 5, 6, 7, 8, or 9, then add 1 to the underlined
digit.
Step 3: Change all the digits after the underlined digit to zeros. If any of the digits after the
underlined digit follow a decimal point, then drop them from the number.
Step 4: Remove the underline from your answer.
We use the sign , which means “is approximately equal to” to compare a
rounded number.
For example: 38 942 39 000.
Example 2
Round the following to the nearest tenth.
a. 13.57
b. 0.034
c. 18.962 487
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Solution
a. Step 1: the digit in the tenths place is a 5. Underline it.
13.57
Step 2: the digit after the 5 is a 7. Add 1 to the 5.
13.67
Step 3: change the 7 to a zero. Since this zero comes after the decimal point, drop it.
13.60 = 13.6
Step 4: 13.57 13.6
b. Step 1: the digit in the tenths place is a 0. Underline it.
0.034
Step 2: the digit after the 0 is a 3. The 0 remains unchanged.
0.034
Step 3: replace the 3 and 4 with zeros and drop them.
0.000 = 0.0
Step 4: 0.034 0.0
c. Step 1: the digit in the tenths place is a 9. Underline it.
18.962 487
Step 2: the digit after the 9 is a 6. Adding 1 to 9 = 10, which is carried to the ones place.
19.062 487
Step 3: replace all the numbers after the underlined number with zeros.
19.000 000
Step 4: 18.962 487 19.0
Now complete Exercise 3 and check your answers
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Exercise 3
1.
2.
Round to the nearest hundred.
a. 8 355
d. 47
b. 203
e. 1 342 969
c. 85
f. 433.789
Round to the nearest one (also called rounding to the nearest whole number).
a. 6.3
b. 18.7
c. 49.6
d. 55.55
e. 0.807
3.
Round as indicated.
a. 43 826 984 to the nearest ten thousand
b. 0.0352 to the nearest tenth
c. 0.499 to the nearest hundredth
d. 0.438 206 to the nearest ten thousandth
e. 0.83 to the nearest whole number
f. 645.96 to the nearest ten
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4.
Round to the nearest hundredth.
a. 0.909
b. 16.583
c. 0.123 456
d. 968.155
e. 0.000 016
5.
Mike had $19.50, Mary had $84.05 and Sandra had $43.88. Round their amounts to
the nearest dollar.
Answers are on page 40.
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Section 4: Operations with decimals
When adding or subtracting whole numbers of decimal fractions it is important that we add
only ones digits to ones digits, tens digits to tens digits, hundreds digits to hundreds digits,
tenths digits to tenths digits and so on. To accomplish this, always make sure that the
decimal points are lined in a column and each decimal fraction has the same number of digits
after the decimal point.
Example 1
Add the following:
1.56 + 38 + 0.009 + 0.7
Solution
Arrange the numbers as shown. Note that a decimal point is included in the number 38 and
every number has been written with three digits after the decimal point.
1.560
38.000
0.009
0.700
40.269
Example 2
Subtract the following:
54 - 9.89
Solution
Align the decimal points. Rewrite 54 as 54.00. Notice that we must borrow from the 4 and
again from the 5.
54.00
-9.89
When numbers are added, the answer is called a sum. The sum of 8 and 5 is 13. When
numbers are subtracted, the answer is called a difference. The difference of 10 and 4 is 6.
Now complete Exercise 4 and check your answers.
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Exercise 4
1.
Find the sum.
a. 846
+95
2.
b. 13
74
812
+195
c.
692
5
84
1387
+42966
Find the sum.
a. 64 + 31 + 942 + 7
b. 88 943 + 369 + 1 003
c. 109 864 + 9 980 + 49 877 + 4 698
3.
Find the sum.
a.
4.
84.26
12.09
b.
16.5
0.03
47.2
92.06
c.
1249.0
70.16
8.297
Find the sum.
a. 84 + 1.2 + 0.96 + 107
b. 29.093 + 14.08 + 9 635
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5.
6.
Find the difference.
a.
182
39
b.
4 602
4 583
c.
116 058
37 069
d.
27.03
5.84
e.
0.029
0.008 43
f.
96
0.74
g.
11.023
7.2
h.
69 003
8 527.81
i.
0.006 837
0.000 938
Find the difference.
a. 83 - 0.83
b. 29 - 6.52
c. 0.5 - 0.003
d. 74.3 - 8.930 2
7.
Mom found three dimes, three quarters, one nickel, seven pennies and two loonies in
Billy’s pocket. How much money did Billy have?
8.
Nadine had $50 before she bought a calculator for $19.95 and a geometry set for
$2.98. How much did she have after these purchases?
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9.
Calculate the balance values.
Date
June 1
10.
Item
June 4
cash withdrawal
June 8
salary deposit
June 12
BC Tel
June 20
June 30
Withdrawal
Deposit
200.00
Balance
480.51
a
891.45
b
33.69
c
car repair cheque
146.35
d
rent cheque
450.00
e
Round each of the numbers to the nearest ten and estimate the sum or difference.
a. 462
95.6
0.25
+79.1
b. 8 299
-315
Answers are on page 41.
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Section 5: Multiplying whole numbers and decimal
fractions
When two numbers are multiplied together, the number is called a product. The numbers
being multiplied together are called factors. For example, when 2 3 = 6, we say that 2 and
3 are factors while 6 is the product.
Example 1
Find the product of 537 and 274.
Solution
Multiply 537 first by 4, then by 7 and then by 2 as shown. Add these products.
537
274
2148
3759
1074
147138
When multiplying decimal fractions, great care must be taken in where to place the decimal
point in the product.
Example 2
Multiply 1.35 by 0.409.
Solution
As above, multiply 1.35 first by 9, then by 0 and then by 4. Arrange these products as shown.
Since the factors have a total of five decimal places, the product must also have five decimal
places.
1.35
0.409
1215
000
540
0.55215
2 decimal places
3 decimal places
5 decimal places
Now complete Exercise 5 and check your answers.
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Exercise 5
1.
Complete the times table chart.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2.
Use the times table chart to help answer the following.
a. the product of 0 and any number is always
.
b. the product of 1 and any number is always
.
c. The product of 5 and any number always ends in a
or a
.
d. Look at the products in the 9’s row or column. Except for the 0 product, the sum
of the digits in each product is always
.
3.
Multiply
a. 41
5
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b. 96
13
c. 944
87
Decimal Fractions
d. 5062
804
27
e.
4.
83
1.1
f.
152
0.07
g.
602
0.504
h. 9065
1.83
Find the products.
a. 17.7 0.6
b. 3.004 0.008
c. 0.006 0.02
d. 0.079 0.000 1
e. 0.008 0.008
5.
Tim earns $1408.45 bi-weekly. There are 26 bi-weekly paydays in one year. How
much will he earn in a year?
6.
Holly drives an average speed of 84 km/hr in her car. How far will she travel in 4.5
hours?
Answers are on page 41.
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Section 6: Dividing whole numbers and decimal
fractions
When one number is divided by another, the answer is called a quotient. The number being
divided is called a dividend, while the dividing number is called a divisor. There is one
number that can never be divided into another. It is impossible to divide zero into any
number. For example, 5 0 is impossible. We say that division by zero is undefined.
There are three ways of indicating division:
quotient
divisor dividend
dividend divisor = quotient
dividend
= quotient
divisor
For example, the problem of dividing 10 by 2 can be written as:
2 10 or 10 2 or
10
2
We read 2 10 as “2 into 10”, 10 2 as “10 divided by 2” and
10
as “10 over 2”.
2
Example 1
Find 860 12 by long division.
Solution
Write the question as 12 into 860. Now, 12 into 8? There are no 12’s in 8. 12 into 86 goes 7
times. Place 7 above the 6 and 7 12 = 84. Place 84 below 86 and subtract. Bring down the
0. Now 12 goes into 20 once. Place the 1 above the 0 and 1 12 = 12. Place 12 below 20
and subtract. Since there is nothing to bring down, 8 is the remainder. We write the quotient
as 71 R8.
71
12 860
84
20
12
8
860 12 = 71 R8
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Whenever we divide whole numbers we often get a remainder. We use the letter R to
indicate the remainder.
When dividing decimal fractions, we must take special care in dealing with the decimal
points. We can make the task simpler by rearranging the question so that we always divide
by a whole number.
Example 2
Find 9.407 0.23 by long division.
Solution
Write as 0.23 into 9.407. Multiply both the divisor and dividend by 100 and rewrite as 23
into 940.7. This allows us to divide by a whole number. Place the decimal point in the
quotient directly above the decimal point in the dividend. Divide as in Example 1. Notice
that 23 goes into 20 zero times.
0.23 9.407
40.9
23 940.7
92
20
0
207
207
0
9.407 0.23 = 40.9
The process of long division can sometimes produce endless non-zero remainders. We can
deal with this by rounding the quotient to an appropriate place value.
Example 3
Find 13 0.7 by long division. Round the answer to the nearest hundredth.
Solution
Write as 0.7 into 13. Multiply both the divisor and dividend by 10. Rewrite as 7 into 130.
Now we can divide by a whole number. Since we must round the quotient to two places after
the decimal point, write three zeros after the 130. Divide as before.
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30
0.7 13
7 130
18.571
7 130.000
7
60
56
40
35
50
49
10
7
3
Rounding the quotient, 13 0.7 = 18.57.
Now complete Exercise 6 and check your answers.
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Exercise 6
1.
Divide.
a. 56 7
b.
54
9
c. 72 8
d. 83 1
e.
0
6
f. 6 0
g.
63
9
h. 43 43
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i. 850 10
j. 6.3 10
k. 94.7 100
l. 0.3 0.03
2.
Divide.
a. 5 124
b. 7 7650
c. 12 6012
d. 9 111.06
e. 41 125.46
f. .08 112
g. 0.06 96
h. 15
. 47.55
i. 0.22 0.6666
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3.
Divide. Round the answers to the nearest hundredths.
a. 3 61
b. 7 20
c. 0.6 11
d. 11 6
e. 1.3 5.4
f. 017
. 9.2
g. 0.41 18.63
h. 23 842 .96
4.
Three children decide to share the cost of their mother’s $40 gift. How much should
each pay? Round the answer to the nearest cent.
5.
Kim drove 310 km on 28.5 litres of gas. How far does she travel on one litre of gas?
Round the answer to the nearest tenth.
6.
Tracy bought 9 apples for $1.71. How much did each apple cost?
Answers are on page 42.
MY MATH PATH 4
Decimal Fractions
34
Section 7: Order of operations
Consider the following calculation:
2+34
Is it 5 4 = 20 or 2 + 12 = 14? In other words, do we add first or multiply first?
Mathematicians have decided upon the following order of operations:
1.
Brackets, such as ( ), { } or [ ]
First, look for brackets and perform all operations within the brackets.
2.
Multiplying and dividing
Second, perform all multiplying and dividing going from left to right.
3.
Adding and subtracting
The last operations to be performed are adding and subtracting, from left to right.
An easier way of remembering the order of operations is using the
acronym BEDMAS. It will help you to remember the order of
operations. The letters stand for:
Brackets or parenthesis
Exponents
Division
Multiplication
Addition
Subtraction
}left to right
}left to right
This is an internationally agreed-upon system for dealing with multiple
operations.
We can now answer the above question:
2 + 3 4 = 2 + 12 = 14
Example 1
Find 50 - 2(3 + 8)
MY MATH PATH 4
Decimal Fractions
35
Solution
First, add 3 + 8 inside the brackets.
50 - 2 (3 + 8) = 50 - 2(11)
Next, multiply 2(11). This expression means 2 11. The operation of multiplication is
always indicated when a number is placed in front of brackets.
50 - 2(11) = 50 - 22
Last, subtract 50 - 22.
50 - 22 = 28
When working towards a solution, it is safest to perform just one operation at a time.
Example 2
Calculate 7 + 3 0.5 10
Solution
First the division, then the multiplication and finally the adding.
7 + 3 0.5 10
= 7 + 6 10
= 7 + 60
= 67
Example 3
Calculate 2[1.4 0.7 0.5 (0.05 + 0.2)]
Solution
First, the innermost brackets. Then, within brackets do multiplying or dividing going from
left to right. Finally, multiply by 2.
2[1.4 0.7 0.5 (0.05 + 0.2)]
= 2[1.4 0.7 0.5 (0.25)]
= 2[2 0.5 (0.25)]
= 2[1.0 (0.25)]
= 2[4]
=8
Now complete Exercise 7 and check your answers.
MY MATH PATH 4
Decimal Fractions
36
Exercise 7
1.
2.
Calculate
a. 12 + 6 3
b. 36 9 2
c. 36 2 9
d. 10 5 + 5
e. 2(8.1 - 0.6)
f. 99 (10 - 0.1)
Calculate
a. 16 - (2 + 3 - 1) 2
b. 18 3 + 3 5 - 2
c. 3[0.3 + 3(3 + 0.3)]
d. 14 - 7 (0.3 + 0.4) 0.2
e. (9 - 7)(9 + 7)
f. 0.5(1.0 - 0.1)(1.0 + 0.1)
g. 4 + 6(8.1 - 1.1) - 25
h. 20 - 12 2 6(0.8 - 0.3)
3.
Donna has saved $180. She plans to save $55 every week for the next 8 weeks. How
much will she have saved altogether after 8 weeks?
4.
Four people went to dinner. Three people had the $12.95 dinner special while the
fourth person had a $14.95 dinner. The wine totaled $22.50. If they split the bill four
ways, how much should each pay?
Answers are on page 42.
MY MATH PATH 4
Decimal Fractions
37
Practice Test
1.
Write the place value for each underlined digit.
a. 84 926
______________
b. 6.0204
2.
Underline the tenths digit in 43.09.
3.
Use >, < or = to compare the numbers below.
4.
a. 93.0 _
90.3
b. 84 _
84.00
c. 0.002 _
0.02
d. 99.9 _
10.000
______________
Round each of the following as indicated.
a. 3 452 to the nearest hundred
b. 16.97 to the nearest tenth
c. 804.99 to the nearest ten
5.
Add 30.7 + 12 + 960.9 + 84.63
6.
Subtract 689.49 from 1 207.3
7.
Multiply the following.
a. 24.06
0.58
MY MATH PATH 4
b.
768
0.097
Decimal Fractions
38
8.
Divide the following.
a. 8 3205
9.
10.
Divide 7 by 0.11 and round the answer to the nearest hundredth.
Calculate the following.
a. 28 - 12 3 2
11.
b. 1.3 0.052
b. 0.4[10 - 0.2(4.8 - 0.8)]
Donna won $5 000 and decided to share half of it among her three children. How
much should each child receive?
MY MATH PATH 4
Decimal Fractions
39
Answers
Introductory Exercise One - page 9
b. .1
Introductory Exercise Two - page 9
b. 5/10 five tenth
one tenth
c. .9
nine tenth
c. .7
seven tenth
Exercise 1 - page 12
1. a. ones
d. ten thousand
g. tenths
j. thousands
b. thousandths
e. hundred thousandths
h. ones
2. a. 842.56
b. 842.56
e. 63 521.009 842
3. a. 823 = 800 + 20 + 3
c. 9 605 = 9 000 + 600 + 5
4.
5.
6.
7.
8.
c. 68 923.05
c. hundredths
f. millions
i. hundredths
d. 843 907 000
2
3
+
10 100
0
8
7
d. 0.087 =
+
+
10 100 1000
b. 8.23 = 8 +
$638
$50.32
$11.04
2 hundred dollar bills, 0 ten dollar bills, 3 loonies, 4 dimes and 1 penny
18 300 000 dollars
Exercise 2 - page 16
1. a. <
b. =
c. >
d. <
e. =
2. a. 1.6
b. 0.040 18
c. 8.07
d. 13 265
e. 0.999 999 999
3. a.
b.
c. =
d.
e.
f.
4. Neither. 49.9 cents = $0.499
Exercise 3 - page 20
1. a. 8400
f. 400
2. a. 6
b. 200
c. 100
d. 0
e. 1 343 000
b. 19
c. 50
d. 56
e. 1
3. a. 43 830 000
4. a. 0.91
b. 0.0
b. 16.58
c. 0.50
c. 0.12
d. 0.4382
d. 968.16
e. 1
e. 0.00
f. 650
5. Mike $20, Mary $84 and Sandra $44
MY MATH PATH 4
Decimal Fractions
40
Exercise 4 - pages 23-25
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
a. 941
a. 1044
a. 96.35
a. 193.16
a. 143
f. 95.26
a. 82.17
$3.17
$27.07
a. $280.51
a. 460
100
0
80
640
b.
b.
b.
b.
b.
g.
b.
1 094
90 315
155.79
9 678.173
19
3.823
22.48
c. 45 134
c. 174 419
c. 1 327.457
c. 78 989
d. 21.19
e. 0.020 57
h. 60 475.19 i. 0.005 899
c. 0.497
d. 65.369 8
b. $1 171.96 c. $1 138.27 d. $991.92
b. 8300
-320
7980
e. $541.92
Exercise 5 - pages 27-28
1.
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10
12
14
16
18
3
0
3
6
9
12
15
18
21
24
27
4
0
4
8
12
16
20
24
28
32
36
5
0
5
10
15
20
25
30
35
40
45
6
0
6
12
18
24
30
36
42
48
54
7
0
7
14
21
28
35
42
49
56
63
8
0
8
16
24
32
40
48
56
64
72
9
0
9
18
27
36
45
54
63
72
81
2. a. 0
b. that number
c. 0 or 5
d. 9
3. a. 205
e. 91.3
b. 1248
f. 10.64
c. 82 128
g. 303.408
d. 4 069 848
h. 165 88.95
4. a. 10.62
e. 0.000 064
5. $36 619.70
6. 378 km
b. 0.024 032
c. 0.000 12
d. 0.000 007 9
MY MATH PATH 4
Decimal Fractions
41
Exercise 6 - pages 32-34
1. a. 8
b. 6
c. 9
d. 83
e. 0
f. Undefined. It is impossible to divide any number by 0.
g. 7
h. 1
i. 85
j. 0.63
k. 0.947
l. 10
2. a. 24 R4 b. 1 092 R6
g. 1 600 h. 31.7
c. 501
i. 3.03
d. 12.34
e. 3.06
f. 140
3. a. 20.33 b. 2.86
g. 45.44 h. 36.65
c. 18.33
d. 0.55
e. 4.15
f. 54.12
c. 8
d. 7
e. 15
f. 10
e. 32
f. 0.495
4. $13.33
5. 10.9 km
6. $0.19
Exercise 7 - page 37
1. a. 14
b. 8
2. a. 14
b. 19
c. 30.6
d. 12
g. 21
h. 2
3. $180 + 8($55) = $620
4. [12.95(3) + 14.95 + 22.50] 4 = $19.08
Practice Test - pages 381.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
a. thousands
b.
43.09
a. >
b. = c.
a. 3500 b. 17.0
1 088.23
517.81
a. 13.9548
b.
a. 400 R5
b.
63.64
a. 20
b. 3.68
$833.3
MY MATH PATH 4
hundredths
<
d. >
c. 800
74.496
0.04
Decimal Fractions
42
Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
MY MATH PATH 4
Moodle Course Developers – Capilano University Students
in the College and University Preparation Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
Decimal Fractions
43
MY MATH PATH 5
COMMON FRACTIONS
Contents
Learning outcomes .....................................................................2
Glossary .....................................................................................2
Unit 1: Operations with fractions ..............................................3
Exercise 1 .........................................................................5
Unit 2: Equivalent fractions ......................................................7
Exercise 2 .........................................................................9
Unit 3: Adding and subtracting common fractions .................11
Exercise 3 .......................................................................13
Unit 4: Multiplying common fractions ...................................16
Exercise 4 .......................................................................17
Unit 5: Dividing common fractions ........................................19
Exercise 5 .......................................................................20
Unit 6: Conversions between common fractions and
decimal fractions ............................................................22
Exercise 6 .......................................................................23
Unit 7: Conversions between decimal fractions and
common fractions ...........................................................25
Exercise 7 .......................................................................26
Unit 8: Estimation ...................................................................28
Exercise 8 .......................................................................29
Practice Test .............................................................................30
Answers....................................................................................32
MY MATH PATH 5
Common Fractions
1
Learning outcomes
Being able to work with whole numbers, decimals and fractions is important in everyday life.
Estimating an answer to a problem helps put the problem in context. This process should be
part of any problem solving activity. Checking answers and solutions to problems is equally
important. The calculator is a useful tool, not only for obtaining accurate results, but also for
checking answers.
When you have completed this Module, you should be able to:
calculate and solve problems involving adding, subtracting, multiplying and dividing
fractions
estimate answers to problems
check that answers and solutions to problems are reasonable in the context of the given
questions
Glossary
Factor
A number that divides evenly into another number. E.g., 2 and 3 are factors of 6.
Improper fraction
An improper fraction has a numerator larger than the denominator.
Mixed number
A mixed number is a combination of a whole number and a proper fraction.
Product
The answer to a multiplication question is called the product.
Proper fraction
A proper fraction has a numerator smaller than the denominator.
Quotient
The answer to a division question is called the quotient.
Sum
The answer to an addition question is called the sum.
MY MATH PATH 5
Common Fractions
2
Unit 1: Operations with fractions
Common fractions are numbers that express parts of a whole. Suppose a pie is cut into five
2
equal pieces and we eat two of them. Then we can say that we ate two-fifths or
of the pie.
5
A common fraction is always made of two numbers. The top number
is called the numerator and the bottom number is called the
denominator.
2 n
u
m
e
r
a
t
o
r
5 d
e
n
o
m
i
n
a
t
o
r
When working with common fractions it is important to remember that the whole is the same
as the number one. If we ate the whole pie, it is the same as eating 5/5 of the pie. Fractions
can be used to represent numbers less than one, more than one or equal to one.
Proper fractions are less than one. Their numerators (top) are less than their denominators.
1
3
9
1
For example,
< 1,
< 1,
< 1 and
< 1.
2
4
10
40
Improper fractions are equal to or greater than one. For example,
and
13
> 1.
12
4
10
9
> 1,
= 1,
>1
3
4
10
Mixed numbers are composed of a whole number and a proper fraction. Mixed numbers are
1
3
1
4
always greater than one. For example, 1 > 1, 5 > 1, 10 > 1 and 2 > 1.
4
8
2
7
The rectangles shown below have been divided into four equal parts. The shaded part can be
9
1
described by the improper fraction as
or by the mixed number 2 .
4
4
MY MATH PATH 5
Common Fractions
3
Example 1
Write
13
as a mixed number.
5
Solution
The idea is to find the number of
5
13
in
or
5
5
13 5
5
3
3
3
= + + =1+1+ =2
5
5
5
5
5
5
The quickest way to write an improper fraction as a mixed fraction is to divide the numerator
by the denominator.
2
13
3
5 13 2
5
5
10
3
Example 2
Write 4
3
as an improper fraction.
8
Solution
The idea is to find the number of eighths in 43/8.
4
3
3 8
8
8
8
3
35
=1+1+1+1+ = + + + + =
8
8
8
8
8 8
8
8
The quickest way to change a mixed number to an improper fraction is to multiply the whole
number by the denominator and add this product to the numerator.
4
3
4 8 3 32 3 35
=
8
8
8
8
MY MATH PATH 5
Common Fractions
4
Now complete Exercise 1 and check your answers.
Exercise 1
1.
Write the fraction that describes the shaded regions.
a.
b.
c.
d.
2.
Write as mixed numbers.
a.
7
5
b.
19
2
c.
41
9
d.
12
4
e.
8
8
f.
64
11
g.
43
22
h.
10
9
i.
84
7
MY MATH PATH 5
Common Fractions
5
3.
Write as improper fractions.
a. 2
1
2
b. 10
c. 4
3
5
d. 8
e. 33
1
3
f. 9
5
8
g. 14
9
10
h. 1
73
100
i. 15
4.
3
4
7
12
Bill has 31 quarters. How much money, in dollars and quarters is this?
Answers are on page 32.
MY MATH PATH 5
Common Fractions
6
Unit 2: Equivalent fractions
In each of the following, one half of the pie is shaded.
1
2
2
4
4
8
1 2
4
, , and
are called equivalent fractions because they represent
2 4
8
1
2
4
the same value. Here,
= = .
2
4
8
The common fractions
There are many, actually infinitely many, ways of expressing the value ½.
1
2
3
4
5
6
7
= = = =
=
=
= ...
2
4
6
8
10 12 14
For the ease of understanding, it is always best to express fractions in the lowest terms. A
fraction is said to be in the lowest terms when neither the numerator or the denominator is
10
divisible by the same prime number. For example,
is not in the lowest terms since both
15
10 and 15 are divisible by 5.
Prime numbers are divisible only by one or themselves:
2, 3, 5, 7, 11, 13, 17, 19, 23, ...
Example 1
Write 140 as a product of primes.
Solution
The process is to keep dividing by the prime numbers 2, 3, 5, 7, 11, ...
Divide 140 by 2.
140 = 2 70
Divide 70 by 2.
140 = 2 2 35
MY MATH PATH 5
Common Fractions
7
35 is not divisible by 2 or 3, so divide 35 by 5.
140 = 2 2 5 7
The quotient 7 is a prime so we are done.
The method used in Example 1 is not the only way to write 140 as a product of primes. We
could easily have obtained the same answer as follows:
140 = 14 10 = 2 7 2 5 = 2 2 5 7
Factoring the numerator and denominator of a fraction is one way of helping to reduce the
fraction to lowest terms. Prime numbers that occur in both the numerator and denominator
can then be cancelled and replaced with a one. For example:
15
3 5
3 5
3
3
20 2 2 5 2 2 5 2 2 4
Example 2
Reduce:
a.
30
63
b.
66
24
c. 5
6
42
Solution
Use the method in Example 1 to factor the numerator and denominator. Replace crossed out
factors with ones. Then multiply the remaining factors.
30 2 3 5 2 3 1 5 10
a.
63 3 3 7 3 1 3 7 21
66
2 3 11
2 1 3 1 11 11
b.
24 2 2 2 3 2 1 2 2 3 1 4
c. 5
6
23
2 1 3 1
1
5
5
5
42
2 3 7
2 1 31 7
7
There are many short cuts to reducing fractions and you may already be aware of some. If
you have a method that works, keep using it. Check with your instructor on the validity of
your method.
Now complete Exercise 2 and check your answers.
MY MATH PATH 5
Common Fractions
8
Exercise 2
1.
Factor (write as a product of primes).
Example: 60 = 2 30 = 2 2 15 = 2 2 3 5
a. 80
b. 18
c. 35
d. 49
e. 77
f. 100
g. 144
h. 20
i. 200
j. 150
2.
List the prime numbers that are greater than 20 but less than 40.
3.
Factor the numerator. Factor the denominator. Then reduce to lowest terms by
cancelling.
a.
9
24
b.
3
6
c.
8
12
d.
10
15
e.
21
21
f.
25
20
MY MATH PATH 5
Common Fractions
9
g. 6
4.
10
35
h.
32
64
26
20
i.
60
70
j.
k.
32
24
l. 12
m.
8
80
n.
o.
26
78
p. 9
18
40
51
17
88
110
Is 2001 a prime number?
Answers are on page 32.
MY MATH PATH 5
Common Fractions
10
Unit 3: Adding and subtracting common fractions
Notice how we can add the shaded parts of the whole.
2
8
3
8
2
3
and have a common denominator. Before adding or subtracting fractions,
8
8
the fractions must be written with a common denominator. The idea behind adding fractions
1
1
that do not have a common denominator, like and
is shown below.
3
4
1
3
1 1 4 4
3 3 4 12
1 1 3 3
+
4 4 3 12
7
1
12
4
The fractions
Notice that the method for writing
lowest terms.
1
4
as
is the reverse method of reducing fractions to
3
12
Example 1
Add
5 3
6 8
Solution
We need a common denominator. Think as follows. The larger denominator is 8. Will 6
divide evenly into 8? No. Now double the 8. 8 2 = 16. Will 6 divide evenly into 16? No.
Triple 8. 8 3 = 24. Will 6 divide evenly into 24? Yes. The common denominator is 24.
Multiply
number.
5 4
3 3
and to get equivalent fractions. Rewrite the answer as a mixed
8 3
6 4
MY MATH PATH 5
Common Fractions
11
5 5 4 20
6 6 4 24
3 3 3 9
+
8 8 3 24
29
24
5
29
1
24
24
Example 2
Subtract 9
4
3
6
5
10
Solution
The common denominator is 10. Subtract the fractions and the whole numbers. Reduce the
answer to the lowest terms.
4
42
8
9
9
5
5 2
10
3
3
3
-6 6 6
10
10
10
5
3
10
9
3
5
1
3
10
2
Example 3
1
1
Subtract 4 1
3
2
Solution
The common denominator is 6. Notice that we cannot subtract
a whole
3
2
from . We must borrow
6
6
6
6
2
from the 4. Then add this to . Now we can subtract.
6
6
6
MY MATH PATH 5
Common Fractions
12
1
1 2
2
4 4
4
6
3
3 2
1
1 3
3
1 1
1
2
23
6
2
6 2
8
3 3
6
6 6
6
3
3
1
1
6
6
5
2
6
4
Now complete Exercise 3 and check your answers.
Exercise 3
1.
Add or subtract as indicated. Reduce your answers to the lowest terms.
3
a. 4
1
4
5
b. 8
3
8
5
6
d.
1
3
6
13
16
5
e. 5
16
9
7
16
4
MY MATH PATH 5
1
5
3
c.
5
4
5
2
Common Fractions
f.
1
3
2
6
3
13
13
g.
4
2
2.
1
2
4
9
h.
6
3
10
i. 16
3
7
12
5
12
Add, then reduce your answers to the lowest terms.
1
4
a.
1
2
2
3
b.
1
4
7
8
c.
3
4
2
15
d.
7
10
3
4
e.
5
24
6
2
3
f.
11
12
1
2
1
g.
3
1
4
2
5
1
h. 17
3
5
6
6
MY MATH PATH 5
16
2
4
Common Fractions
14
3.
Subtract, then reduce your answers to lowest terms.
5
6
a.
1
3
13
d.
4
3
4
3
16
g.
2
5
3
5.
6.
5
8
c.
1
2
4
1
2
e.
2
11
3
3
8
f.
3
4
19
7
5
7
10
h.
3
21
4
10
4.
59
100
b.
3
10
22
If Jack ate half a pie and Jill ate a third of the same pie, how much was left over?
1
1
1
hours on Monday, 5 hours on Wednesday and 4 hours on
4
2
3
Friday. How many hours altogether did he work?
Roy worked 3
Joan cut 7
3
inches off a 12 inch board. How much of the board was left?
4
Answers are on page 33.
MY MATH PATH 5
Common Fractions
15
Unit 4: Multiplying common fractions
The pie below is cut into 8 parts.
or
3
1
3
of the pie is shaded. Suppose we want to find of
4
2
4
1 3
. Notice that the word “of ” means “times” in arithmetic.
2 4
3
4
1
3
1 3
3
of = =
2
4
2 4
8
When multiplying fractions, multiply the numerator times the numerator and the
denominator times the denominator.
Example 1
Multiply
2
4
3
5
Solution
Simply multiply numerators and denominators.
2
4
24 8
=
3
5
3 5 15
Example 2
Multiply 2
3
5
4
Solution
Write both numbers as improper fractions, then multiply numerators and denominators.
Rewrite the answer as a mixed number.
2
11 5 11 5 55
3
3
5
13
4
4 1 41 4
4
It is sometimes possible to cancel common factors from the question before multiplying.
MY MATH PATH 5
Common Fractions
16
Example 3
Multiply
4
1
3
2
21
Solution
1
7
as . Factor all numerators and denominators. Cancel common factors. If we
2
2
4 7 28
28
and we would have to reduce
to lowest terms
did not cancel first, then
21 2 42
42
regardless.
Write 3
4
1 4 7 2 2 7 2 1 2 7 1 2
3
21
2 21 2 3 7 2 3 7 1 2 1 3
The steps to multiplying fractions are:
Step 1: Rewrite any mixed numbers as improper fractions.
Step 2: Factor, if possible, all numerators and denominators.
Step 3: Cancel factors common to both the numerator and denominator.
Step 4: Multiply the remaining factors.
Now complete Exercise 4 and check your answers.
Exercise 4
1.
Multiply, then write the products in the lowest terms.
a.
2 7
3 11
b.
5 3
12 5
c.
5 8
8 5
d.
7
1
1
8
2
MY MATH PATH 5
Common Fractions
17
1
e. 3 5
4
g. 12
i. 2
2.
3
4
Shelly exercised
4.
Bill did
7
1
5
3
10
1
3
h. 1 1
2
5
7
1
3
8
5
exercise?
3.
f. 2
j.
3
2
7
4
9
3
of an hour every day. How many hours in a week did she
4
2
of the 69 math questions. How many questions did he not do?
3
1
hour long English class staring out the window.
2
How much time did she spend doing this?
Carol spent three quarters of the 1
Answers are on page 33.
MY MATH PATH 5
Common Fractions
18
Unit 5: Dividing common fractions
How many quarters are there in one half? The drawing suggests that there are two quarters in
the shaded half of the pie. Mathematically, we are asking, what is one half divided by one
quarter, or
1
2
1 1
2
2 4
1 4
1
2 . The number 4 is the reciprocal of . The reciprocal of a
2 1
4
fraction is found by interchanging the numerator and the denominator. Note that your
1
calculator has a reciprocal key
x
Notice also that
Division questions can be rewritten as multiplication questions. Dividing by a number is the
same as multiplying by the reciprocal of that number.
Example 1
Divide
1 2
2 3
Solution
Rewrite as a multiplication question by changing the “” to a “” and the
3
, then multiply.
2
2
to its reciprocal
3
1 2 1 3 1 3 3
2 3 2 2 22 4
MY MATH PATH 5
Common Fractions
19
Example 2
3
Divide 6 5
8
Solution
Rewrite as improper fractions. Then rewrite as a multiplication question. Then multiply and
rewrite the answer as a mixed number.
3
51 5 51 1 511 51
11
6 5
1
8
8 1 8 5 8 5 40
40
Now complete Exercise 5 and check your answers.
Exercise 5
1.
Divide
a.
1 3
2 5
b.
1 1
2 2
c.
3
2
4
d.
7 4
12 5
f.
1
1
2
8
4
1 3
e. 1
2 4
g. 12 2
2
5
MY MATH PATH 5
2
1
h. 3 5
3
4
Common Fractions
20
i. 3
1
2
j. 4
1
7
3
8
2
1
hours. How much time did it take her to make one plate?
4
2.
Kim made 9 plates in 2
3.
Greg cut two thirds of the cake into five equal pieces and then ate one piece. How
much of the whole cake did Greg eat?
Answers are on page 33.
MY MATH PATH 5
Common Fractions
21
Unit 6: Conversions between common fractions
and decimal fractions
It is often convenient to write a common fraction as a decimal fraction. For example, we
think of three quarters as 75 cents or $0.75. In other words
3
0.75
4
To write a common fraction as a decimal fraction, divide the numerator by the denominator.
3
3 4 or 4 3
4
Example 1
Write
3
as a decimal fraction.
8
Solution
3
as a long division question. Write 3 as 3.00 and add more decimal place zeros as
8
needed.
Write
0.375
8 3.000
24
60
56
40
40
0
3
= 0.375
8
Example 2
MY MATH PATH 5
Common Fractions
22
Write 5
1
as a decimal fraction.
3
Solution
1
. Note that the quotient has repeating threes. Use a
3
bar over the repeating digit. Place the whole number 5 before the decimal. 5.33
Use long division to divide the fraction
.33
3 1.00
1
= 5. 3
3
5
Now complete Exercise 6 and check your answers.
Exercise 6
1.
2.
Rewrite as decimal fractions and then memorize the results.
a.
1
2
b.
1
3
c.
2
3
d.
1
4
e.
2
4
f.
3
4
g.
1
5
h.
2
5
i.
3
5
j.
4
5
Rewrite as decimal fractions.
MY MATH PATH 5
Common Fractions
23
a.
1
6
b.
d.
5
8
e. 2
g.
3
10
h.
j. 6
3.
4.
5
6
1
4
3
100
c.
1
8
f.
1
12
i. 1
5
12
7
8
Rewrite as decimal fractions and note the pattern.
a.
1
9
b.
2
9
c.
3
9
d.
4
9
e.
5
9
f.
6
9
g.
7
9
h.
8
9
i.
9
9
Write as decimal fractions and notice the pattern.
a.
20
99
b.
52
99
c.
1
99
d.
13
99
e.
5
99
f.
98
99
Answers are on page 34.
MY MATH PATH 5
Common Fractions
24
Unit 7: Conversions between decimal fractions and
common fractions
There are three types of decimal fractions:
1.
Terminating decimals have a fixed number of digits. For example, 0.5, 3.625 and
10.128 496.
2.
Repeating decimals have one digit or a group of digits which repeat forever. A bar is
placed over the repeating digit or group of repeating digits. For example:
0.333 333 333... = 0. 3
6.934 934 934... = 6. 934
8.056 262 626 = 8.0562
3.
Non-repeating decimals have an infinite number of digits but no repeating digit or
group of digits. For example:
0.123 456 789 101 112...
6.505 505 550 555 505...
3.141 592 748 605 213...
Both terminating and repeating decimals can be written as common fractions. Non-repeating
decimals cannot be written as common fractions. Study the following:
0.7 =
7
10
0.029 =
0.83 =
29
1000
5.60891 = 5
60891
100000
83
100
0.2011=
2011
10000
0.029 =
29
1000
When a decimal is written as a fraction, the decimal digits appear in the numerator of the
fraction and the denominator has the same number of zeros as there are decimal digits.
Example 1 - Write 8.0306 as a common fraction. Reduce to the lowest term.
Solution
Write as a mixed number. The decimal part of 8.0306 has four digits, so the denominator is
10 000.
8.0306 = 8
MY MATH PATH 5
306
153
8
10000
5000
Common Fractions
28
Now complete Exercise 7 and check your answers.
Exercise 7
1.
Write as common fractions but do not reduce to the lowest terms.
a. 0.5
b. 0.61
c. 0.907
d. 1.1
e. 6.02
f. 0.70
g. 0.001
h. 27.95
i. 0.0105
j. 1.502
2.
Write as common fractions and reduce to the lowest terms.
a. 0.5
b. 0.25
c. 0.125
d. 0.2
e. 0.04
f. 0.75
g. 0.6
h. 0.4
MY MATH PATH 5
Common Fractions
29
3.
Write as decimal fractions.
a.
7
9
b.
14
99
c.
2
9
d.
82
99
Answers are on page 34
MY MATH PATH 5
Common Fractions
30
Section 8: Estimation
Estimation is a useful tool in mathematical problem solving. Estimating gives you an
approximate or “ballpark” answer. People often estimate differently. There is no one correct
way to estimate. It takes practice to get good at choosing numbers that make estimating
easier. Here are some estimating strategies:
1.
Round to numbers that are easy for you to compute. For example, about how much is
29 14? Some possible estimates are:
25 16 = 400
30 10 = 300
30 15 = 450
2.
Use numbers that make sense to you. When you round, sometimes the result will be
an underestimate (less than the exact answer) and sometimes it will be an
overestimate (more than the exact answer).
problem
estimate
31 4
30 4 = 120
48 5
50 5 = 250
17 21
20 20 = 400
3.
the exact answer is...
a little more than 120
a little less than 250
about 400
Choose numbers that are compatible and easy to work with for the problem. For
example, how much is 2775 divided by 6? Some possible estimates are:
3000 6 = 500
2800 7 = 400
4.
Use whole numbers to estimate fractions and decimals. For example:
problem
19.32 - 7.29
3.7 8.2
2
1
16 + 12
3
8
3
1
20 3
4
5
5.
estimate
19 - 7 = 12
4 8 = 32
17 + 12 = 29
21 3 = 7
Group numbers to make estimation easier. For example, 48 + 27 + 55 + 75 + 98 is
approximately = 300. This is because 48 + 55 is about 100, 27 + 75 is about 100 and
98 is about 100.
Now complete Exercise 8 and check your answers.
MY MATH PATH 5
Common Fractions
31
Exercise 8
1.
2.
Decide whether these will total more or less than $20.
a. $7.99 + $8.99
b. $4.79 + $8.39 + $7.19
c. $12.87 + $6.99
d. $4.99 + $3.29 + $6.89
e. $5.89 + $13.99
f. $7.69 + $5.49 + $7.99
Estimate the following
a. $4.99 + $3.29 + $6.89
c. 2
1
1
1
+4 +3
3
8
4
b. 7.4 9.8
d. 4397 - 1970
3
3
- 49
5
4
e. 125 + 98 + 73 + 142 + 59
f. 174
g. 17.4 + 9.1 + 2.8 + 10.7
h. 795 4
i. 9
2
7
4
3
8
j. 8125 42
Answers are on page 34.
Now complete the Practice Test and check your answers.
MY MATH PATH 5
Common Fractions
32
Practice Test
1.
Write as mixed numbers.
a.
2.
25
3
3
5
b. 16
2
3
Reduce to lowest terms.
a.
4.
39
6
Write as improper fractions.
a. 4
3.
b.
25
30
b. 8
12
90
Add or subtract as indicated, then reduce the answers to lowest terms.
2
5
a.
1
3
7
8
b.
1
2
12
d.
3
7
10
MY MATH PATH 5
2
5
5
c. 18
12
1
3
6
3
16
e.
3
4
4
Common Fractions
33
5.
Multiply or divide, then reduce the answers to the lowest terms.
a. 6
c.
6.
9
5
10 12
b. 4
1
1
3 2
d. 3
3
2
2
8
5
b. 3
4
11
Write as decimal fractions.
a.
7.
3
4
5
8
Write as common fractions.
a. 0.501
b. 12. 3
8.
Audrey claims to have read two-thirds of a 396 page report. How many pages did she
read?
9.
Bill did
10.
11.
1
1
of his assignment the first night, then the second night. How much
4
3
would he have left to do on the third night?
3
cup of sugar if you want to serve 6 people. How much sugar
4
should you use to serve 2 people?
The recipe calls for
Estimate the following.
a. 14.7 + 12.36 + 18.25
12.
b. 5
3
1
5
3 9
8
6
4
In driving a distance of 489 km, Paul’s car consumes 51.2 litres of gasoline. Estimate
how many kilometres he can travel on 1 litre of gasoline.
Answers are on pages 35.
MY MATH PATH 5
Common Fractions
34
Answers
Exercise 1 - page 5
1. a.
5
8
2
5
21
g. 1
22
5
3. a.
2
2. a. 1
g.
4.
149
10
10
1
0
or 3
c.
or 0
5
3
3
1
5
b. 9
c. 4
d. 3
2
9
1
h. 1
i. 12
9
43
23
8
b.
c.
d.
1
4
5
b.
h.
173
100
i.
d.
12
or 3
4
e. 1
e.
f. 5
100
3
f.
9
11
77
8
187
12
31
3
= 7 or 7 dollars and 3 quarters
4
4
Exercise 2 - page 9
1. a. 80 = 2 40 = 2 2 20 = 2 2 2 10 = 2 2 2 2 5
b. 18 = 2 9 = 2 3 3
c. 35 = 5 7
d. 49 = 7 7
e. 77 = 7 11
f. 100 = 2 50 = 2 2 25 = 2 2 5 5
g. 144 = 2 72 = 2 2 36 = 2 2 2 18 =2 2 2 2 9 = 2 2 2 2 3 3
h. 20 = 2 10 = 2 2 5
i. 200 = 2 100 = 2 2 2 5 5
j. 150 = 2 75 = 2 3 25 = 2 3 5 5
2. 23, 29, 31, 37
3
1
3. a.
b.
2
8
2
1
g. 6
h.
2
7
1
m.
n. 3
10
2
3
6
i.
7
1
o.
3
c.
2
3
13
j.
10
4
p. 9
5
d.
e. 1
k.
4
3
f.
5
4
l. 12
9
20
4. No. 2001 = 3 23 29
MY MATH PATH 5
Common Fractions
35
Exercise 3 - pages 13
2
1
8
8
3
4
2
27
11
=
b.
=1
c.
or 1
d. 1 = 1
e. 14
= 15
2
8
5
5
6
3
4
16
16
2
1
7
f. 6
g. 2
h. 2
i. 24
3
2
10
3
11
13
5
25
5
19
7
a.
b.
c.
or 1
d.
=
e. 40
= 41
4
8
8
30
6
12
12
12
19
7
13
1
47
17
f. 2
=3
g.
=1
h. 27
= 28
30
30
12
12
12
12
3
1
29
3
1
5
a.
=
b.
c. 5
d. 8
e. 7
6
2
100
8
4
6
5
25
19
f. 4
g. 4
h.
8
48
20
5
1
1 1
1 - = 1 - = of the pie was left over
2 3
6
6
1
1
1
13
1
3 + 5 + 4 = 12
= 13
hours
4
2
3
12
12
3
1
12 - 7 = 4 inches
4
4
1. a.
2.
3.
4.
5.
6.
Exercise 4 - page 17
14
1
b.
c. 1
4
33
2
2
f. 14
g. 9
h. 2
5
5
3
21
1
2.
7=
= 5 hours in a week
4
4
4
2
3. 69 - 69 = 69 - 46 = 23 questions or 1
3
3
1
3
3
9
1
4.
1 =
= or 1 hours
4
2
4
2
8
8
21
5
=1
16
16
1
i. 9
5
1. a.
d.
65
1
= 16
4
4
5
j. 5
12
e.
2
1
69 = 69 = 23 questions
3
3
Exercise 5 - page 20
1. a.
5
6
g. 5
b. 1
h.
MY MATH PATH 5
44
63
c.
3
8
i. 6
35
48
5
j. 1
31
d.
Common Fractions
e. 2
f.
1
18
36
1
9
1
1
9=
= hour per plate
4
4
9
4
2
2
3.
5=
of the cake
3
15
2. 2
Exercise 6 - page 23
1. a.
f.
2. a.
f.
3. a.
f.
4. a.
d.
1
= 0.5
2
3
= 0.75
4
016
.
0.083
1
= 0.1
9
6
= 0. 6
9
20
0.20
99
13
0.13
99
b.
g.
b.
g.
b.
g.
1
2
= 0. 3 c.
= 0. 6
3
3
1
2
= 0.2
h.
= 0.4
5
5
c. 0.125
0.83
0.3
h. 0.03
2
3
= 0. 2 c.
= 0. 3
9
9
7
8
= 0. 7 h.
= 0.8
9
9
52
b.
0.52
99
5
e.
0.05
99
d.
i.
d.
i.
d.
i.
c.
f.
1
= 0.25
4
3
= 0.6
5
0.625
1416
.
4
= 0. 4
9
9
=1
9
1
0.01
99
98
0.98
99
e.
j.
e.
j.
e.
2
= 0.5
4
4
= 0.8
5
2.25
6.875
5
= 0.5
9
Exercise 7 - page 26
1. a.
g.
2. a.
g.
3. a.
5
10
1
1000
1
2
3
5
0. 7
b.
h.
b.
h.
b.
61
100
95
27
100
1
4
2
5
0.14
907
1000
105
i.
10000
1
c.
8
c.
c. 0. 2
1
10
502
j. 1
1000
1
d.
5
d. 1
e. 6
e.
2
100
1
25
f.
70
100
f.
3
4
d. 0.82
Exercise 8 - page 29
1. a. less
b. more
c. less
2. (answers may vary slightly)
a. $15
b. 70
c. 9
g. 40
h. 200
i. 50
MY MATH PATH 5
d. less
e. less
f. more
d. 2400
j. 200
e. 500
f. 125
Common Fractions
37
Practice Test - pages 30
1
1
b. 6
3
2
23
50
b.
5
3
5
2
b. 8
6
15
11
3
b.
8
15
9
1
2
26
or 4
b.
or 8
2
3
2
3
0.625
b. 3. 36
501
1
b. 12
1000
3
1. a. 8
2. a.
3. a.
4. a.
5. a.
6. a.
7. a.
c. 25
c.
3
8
3
20
3
10
45
13
d.
or 1
32
32
d. 8
e. 3
7
16
2
396 = 264 pages
3
7
5
1 1
=
of the assignment
9. 1 - = 1 4 3
12 12
2
3
1
10.
= cup
6
4
4
11. (answers may vary slightly)
a. 45
b. 180
12. 10 kilometres
8.
MY MATH PATH 5
Common Fractions
38
Moodle Courses and Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
MY MATH PATH 5
Moodle Course Developers – Capilano University
Students in the College and University Preparation
Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
Common Fractions
39
MY MATH PATH 6 - MODULE 1
RATIOS AND PROPORTIONS
Contents
Learning outcomes .....................................................................2
Glossary .....................................................................................2
Introduction ................................................................................3
Section 1: Ratios and rates ........................................................4
Exercise 1 .........................................................................7
Section 2: Proportions ...............................................................9
Exercise 2 .......................................................................11
Section 3: Solving proportions................................................12
Exercise 3 .......................................................................14
Section 4: Applications ...........................................................15
Exercise 4 .......................................................................16
Practice Test .............................................................................17
Answers....................................................................................19
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
1
Learning outcomes
Proportions can be used to solve a wide variety of problems, from consumer problems to
trade-related problems. The method for solving proportions will be used in later topics such
as percent, algebra, trigonometry and geometry.
When you have completed this Module, you should be able to:
read, write, interpret, and compare ratios
read, write and identify proportions and use them to solve problems
Glossary
Cross-products
The products you get when you cross-multiply ratios are called cross-products.
Proportion
If two ratios are equal, they form a proportion.
Rate
A comparison of different Sections.
Ratio
A ratio is a comparison of similar Sections.
Variable
The letter used to represent an unknown quantity is a variable.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
2
Introduction
Proportions are very useful in solving problems in everyday life. A proportion is made up of
two equal fractions or ratios.
You can use ratios to compare how much chocolate mix is required for a quantity of milk.
You can find out how much chocolate mix should be used with 1 litre (1000ml) of milk, if
you know that it takes 25 g of chocolate mix to 250 ml of milk for 1 serving.
25g 250ml
x
1000ml
Since
25
250
, we can see that the x in the ratio is 100 grams.
100 1000
So, you would need 100 g of chocolate mix for 1 litre of milk.
In this Module, you will work with ratios that compare 2 or 3 quantities. You will also learn
a method for finding out if two ratios form a proportion and how to find a missing part of a
proportion. You will then use proportions to solve problems.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
3
Section 1: Ratios and rates
Ratios are used frequently in everyday life to compare two or more quantities having the
same Sections. For example, when you prepare a tin of orange juice at a mix of 1 to 3, that
means you use one tin of concentrate to three tins of water.
A ratio is simply a comparison of two or more quantities.
A ratio can be written in any of three ways. The example of the orange juice can be written:
in words:
with a colon:
or as a fraction:
1 to 3
1:3
1
3
In all cases, the ratio is read as: “one to three.”
Ratios are easy to use, but there are a few simple rules:
The quantity that is given first must be written first or must be given as the
numerator of the fraction.
Example 1
Write “three to two” as a ratio.
Solution
3 to 2, 3:2, or
3
2
The ratio should, like fractions, be reduced to its lowest terms.
Example 2
Reduce the ratio of 12:2.
Solution
12:2 = 6:1 or
MY MATH PATH 6
6
6
[Note: the 1 in the ratio
must be included.]
1
1
MODULE 1 – Ratios and Proportions
4
Example 3
Reduce the ratio
75
.
50
Solution
3
75 3 25
2
50 2 25
Example 4
Unlike fractions, however, there may be more than two terms in a ratio. Fertilizer bags
contain 3 numbers like 16:20:24 that identify the ratio of phosphorus to nitrogen to potash.
If the relative amount of each quantity was cut in half, the new ratio would be 8:10:12. Even
though the amounts are less, the ratio between them remains the same. This can even be
reduced further by halving each number again.
Solution
16:20:24 = 8:10:12 = 4:5:6
When comparing measurements, both quantities should normally use the same
Section of measure.
Example 5
Compare “10 m to 100 cm”. You must first convert both measurements to either metres or
centimetres. (Remember, 1 m = 100 cm.)
Solution
Converting both to centimetres:
10 m to 100 cm
becomes
1000 cm to 100 cm or 1000:100
reducing to 10:1
Converting both to metres:
10 m to 100 cm
becomes
10 m to 1 m
or
10:1
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
5
Notice that, regardless of which Section we choose, metres or centimetres, the ratio is the
same and is written without the Sections.
A ratio may also be used to express a rate such as kilometres per hour (km/h) or revolutions
per minute (rpm).
In these cases, since the quantities cannot be converted to similar Sections, the
Sections must be written with the rate.
Example 6
The batter hit 4 pitches out of 8. Write this as a rate.
Solution
4 hits
1 hit
8 pitches 2 pitches
Now complete Exercise 1 and check your answers.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
6
Exercise 1
1.
Write the following as ratios. Use both “:” notation and “fraction” notation.
8
For example: 8 km to 15 km = 8:15 or
15
a. $7 to $10
b. 23 g to 46 g
c. 1 coffee to 4 coffees
d. 10 oranges to 3 oranges
2.
Write the following as ratios: (Remember to reduce them to lowest terms.)
a. 3 cups of flour to 2 cups of sugar
b. 6 cm to 11 cm
c. 10 mm to 12 mm
d. 20 cm to 1 m
e. 3 minutes to 1 hour
f. 3 parts blue to 5 parts green to 7 parts white
g. 15% phosphorus to 20% nitrogen to 15% potash
h. $9.00 to $12.00 to $6.00
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
7
3.
Write the following as rates. Reduce them to lowest terms.
a. Joanne drove 250 km in 5 hours. What was her average speed in km per hour?
b. Martin was paid $480 for working a 40 hour week. How much was he paid per
hour?
c. A baseball player has 24 hits out of 75 times at bat. Write his batting average as a
rate.
Bonus question: Batting averages are usually expressed as a decimal. Convert the
above rate into a decimal.
Answers are on page 19.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
8
Section 2: Proportions
A statement that two ratios are equal is called a proportion. For example,
4 8
6 12
This statement is a proportion because both ratios reduce to
2
3
4 2 2 2
6 3 2 3
8 2 4 2
12 3 4 3
We can also determine if two ratios are equal by using a method called cross-multiplying.
The products are called cross-products.
If the cross-products of two ratios are equal, then the ratios form a proportion.
Example 1
Is
4
8
equal to
?
6
12
Solution
4 12 = 48 and 6 8 = 48
so
4 8
This is a proportion.
6 12
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
9
Example 2
Is 2:5 equal to 10:25?
Solution
Write the ratios as
2
5
and
2
10
and
and cross-multiply.
5
25
10
25
2 25 = 50
5 10 = 50
So
2 10
. This is a proportion.
5 25
Example 3
Is
7
15
equal to
?
8
17
Solution
7
8
and
15
17
7 17 = 119
8 15 = 120
The ratios are not equal and so this is not a proportion.
7 15
( means “not equal to”)
8 17
Now complete Exercise 2 and check your answers.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
10
Exercise 2
1.
Determine if the following ratios are equal.
a. 2:3, 6:9
b.
4 12
,
5 15
c. 20:3, 40:6
d.
15 25
,
8 14
e. 3:4, 8:12
f.
4.5 8.5
,
9 17
g. 7:2, 42:12
h.
i.
0.3 1
,
6.9 22
1
4 1
,
5 4
1
j. 100:8, 25:2
Answers are on page 19.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
11
Section 3: Solving proportions
Whenever we know three quantities in a proportion we can always find the fourth. The most
common method of doing this is called cross-multiplying.
Example 1
2 1
6 n
Find n.
Solution
Cross-multiply.
2n=61
2n = 6
n
6
(to find n divide 6 by 2)
2
n = 3 (missing term)
In the example above, we write 2n as 2n. Whenever a letter is multiplied by a number, there
is no need to write a times sign between them. For example, 13y means 13 times y. The
letters we use to represent unknown quantities in mathematics are called variables.
It is also important to understand the step we took after cross-multiplying in the previous
example. If we know that 2 times n equals 6, or 2n=6, then we can find n by dividing both 2n
and 6 by 2.
Example 2
y 4
21 7
Find y.
Solution
7 y = 4 21
7y = 84
y
84
7
y = 12 (missing term)
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
12
Example 3
1
3 6
x 9
Find x.
Solution
6x=
1
9
3
9
3
6x =
6x = 3
x=
3
6
x=
1
(missing term)
2
Example 4
20 is to 3 as x is to 15
Solution
In this problem we need to first write the proportion and then solve for the unknown variable.
20 x
3 15
3 x = 20 15
3x = 300
x
300
3
x = 100 (missing term)
Now complete Exercise 3 and check your answers.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
13
Exercise 3
1.
Solve each proportion for the letter given.
a.
2 N
3 12
g. 10:3 = 4:c
b.
15 45
1 y
h. 7:d = 1:6
c.
9 5
a 3
i. x:3 =
d.
x 8
5 2
j. 15 is to 1 as 45 is to y
e. x:5 = 8:2
k. 2 is to 3 as N is to 12
f. 2:3 = r:17
l. 9 is to a as 5 is to 3
1
:9
2
Answers are on page 19.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
14
Section 4: Applications
Proportions are easily used to solve word problems.
Example 1
If a dozen apples cost $1.80, how much will 8 apples cost?
Solution
Convert the problem to a proportion: a dozen apples is to $1.80 is the same as 8 apples is to
y, where y is the cost of 8 apples.
apples
12
8
1.80 y
Make sure similar Sections are aligned on
the horizontal like in this example OR
aligned on the verticle like in example
2A. Either way, you will get the same
answer.
price
12 y = 1.80 8
12y = 14.40
y
14.40
12
y = $1.20
Therefore, 8 apples would cost $1.20.
Example 2
If a car travels 400 km on 52 litres of gas, how many litres will it take to travel 250 km?
Solution
400km 52litres
250km xlitres
400 x = 250 52
400x = 13000
x
OR
similar Sections
are aligned on the
VERTICLE.
400km 250km
52litres xlitres
similar Sections
are aligned on the
HORIZONTAL.
13000
400
x = 32.5 L
It will take 32.5 L to travel 250 km.
Now complete Exercise 4 and check your answers.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
15
Exercise 4
Solve each problem. Round answers to the nearest hundredth where appropriate.
1.
4 litres of paint are needed to paint 64 fence posts. How many posts can be painted
with 5 litres of paint?
2.
On a road map 1 cm is used to represent 10 km. How far apart are towns that are 6
cm apart on the map?
3.
A 1.3 m gold wire costs $200. How much will 0.8 m of the wire cost?
4.
An engine uses a 40:1 mixture of gas to oil. How much oil should be added to 5 litres
of gasoline?
5.
3 kg of dates cost $27. How much will 0.5 kg of dates cost?
6.
The label on a medicine bottle indicates that 10 cm3 of liquid contains 250 mg of
medication. A prescription requires 40 mg of medication. How much liquid is
required?
7.
A pharmacist pays $12.65 for 60 medicine capsules. At this rate how much would
150 capsules cost?
8.
If it takes 4 hours to drive 380 km, how long will it take to drive 700 km?
Answers are on page 20.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
16
Practice Test
1.
Write as ratios and reduce to lowest terms:
a. 3 dimes to three quarters
b. 6 minutes to one hour
2.
Reduce these ratios to lowest terms.
a. 12:14
b. 15:18:24
c.
3.
57
19
Determine whether the following ratios are equal:
a. 1:6 and 36:216
b.
4.5 24
,
9 48
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
17
4.
Solve the proportions for the given variable:
a. 9:12 = x:16
b. b:3 = 20:3
c.
10 18
c 45
d.
6 d
18 3
e. 7.2:6 = 18:e
f.
f
85.5
20
90
5.
If 2 kg of grass seed will cover 15 square metres of lawn, how many kg of seed are
required for a lawn of 26.25 square metres?
6.
In a poll of 2000 people, 650 said that they preferred chocolate ice cream, and the rest
said they preferred vanilla. If you were ordering ice cream for 5000 people, how
many would you expect to want vanilla?
Answers are on page 20.
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
18
Answers
Exercise 1 - pages 7-8
1. a. 7:10 or
c. 1:4 or
2.
7
10
1
4
b. 23:46 or
d. 10:3 or
a. 3:2
d. 1:5
g. 3:4:3
3. a. 50 km/hr.
23
46
10
3
b. 6:11
e. 1:20
h. 3:4:2
c. 5:6
. 3:5:7
b. $12/hr.
c. 8/25 = 0.32
Exercise 2 - page 11
1. a. yes
b. yes
c. yes
d. no
g. yes
h. no
i. yes
j. yes
e. no
f. yes
Exercise 3 - page 14
1. a. N = 8
27
2
c. a =
or 5 or 5.4
5
5
e. x = 20
1
g. c = 1 or 1.2
5
1
i. x =
6
k. N = 8
MY MATH PATH 6
b. y = 3
d. x = 20
f. r =
34
1
or 11 or 11.3
3
3
h. d = 42
j. y = 3
l. a =
27
2
or 5 or 5.4
5
5
MODULE 1 – Ratios and Proportions
19
Exercise 4 - page 16
2.
1 6
x = 60 km
10 x
1.3m 0.8
c = $123.08
200 c
4.
40 5
1 x
5.
3 0.5
c = $4.50
27 c
6.
10 x
x = 1.6 cm3
250 40
7.
12.65 c
c = $31.63
60 150
8.
4
x
x = 7.37 h
380 700
1.
4 5
64 x
3.
x = 80 posts
x = 0.125 L or 125 mL
Practice Test - pages 17-18
2
or 2:5
5
2. a. 6:7
1. a.
1
or 1:10
10
b. 5:6:8
b.
3. a. yes
b. yes
4. a. x = 12
b. b = 20
c. c = 25
d. d = 1
e. e = 15
f. f = 19
c. 3:1
5. 3.5 kg
6. 3375 people
MY MATH PATH 6
MODULE 1 – Ratios and Proportions
20
Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
MY MATH PATH 6
Moodle Course Developers – Capilano University
Students in the College and University Preparation
Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
MODULE 1 – Ratios and Proportions
21
MY MATH PATH 6 - MODULE 2 PERCENT
Contents
Learning outcomes .....................................................................2
Glossary .....................................................................................2
Introduction ................................................................................2
Unit 1: Converting between decimals and percents ..................3
Exercise 1 .........................................................................5
Unit 2: Converting between fractions and percents ..................7
Exercise 2 .........................................................................9
Unit 3: Adding and subtracting percents.................................11
Exercise 3 .......................................................................12
Unit 4: Percent increase and decrease .....................................13
Exercise 4 .......................................................................14
Unit 5: Percent applications - sales tax ...................................15
Exercise 5 .......................................................................15
Practice Test .............................................................................16
Answers....................................................................................17
MY MATH PATH 6
Module 2 Percent
1
Learning outcomes
Percent is often used in everyday life. Percents are used to calculate taxes (PST and GST),
markups, discounts, interest on loans, credit cards and bank accounts and nutrient values in
foods.
When you have completed this Module you should be able to:
convert between decimals, fractions and percent
solve a variety of percent problems
Percent
Glossary
An amount out of 100. It is indicated by the symbol %.
Introduction
Percent, or the symbol %, is very common in everyday life. Sales are described as 25% or
50% off, mortgages are lent at 13% interest and sales tax is 7%. The word percent means
“per hundred”. The word “cent” comes from the Latin word centum which means hundred.
The following are several common statements involving percents:
I got 80% on my math test
G.S.T. is 7%
sales tax is 7%
credit card interest rates vary from 15-20%
Percents (or percentages as they are commonly called) are closely related to fractions and
decimals.
7% means 7 per hundred
7
means 7 per hundred
100
0.07 means 7 per hundred, so
7
= 0.07
100
When we change from percent notation to fraction or decimal notation, we are simply
1
replacing the percent sign with
or 0.01.
100
7% =
MY MATH PATH 6
Module 2 Percent
2
Unit 1: Converting between decimals and percents
Percent to decimal
To convert from a percent to a decimal:
1.
Write the numeral without the percent sign.
2.
Multiply the numeral by 0.01; which means you move the decimal
point two places to the left.
Example 1
Convert 7.25%, 136% and 16¾% to decimals.
Solutions
1.
7.25% = 7.25 0.01 = 0.0725 (move the decimal two places to the left)
7.25% = 0.0725
2.
136% = 136 0.01 = 1.36 (move the decimal two places to the left)
3.
3
3
16 % = 16.75 0.01 = 0.1675 (convert the fraction
to the decimal 0.75, then
4
4
move the decimal two places to the left)
MY MATH PATH 6
Module 2 Percent
3
Decimals to percent
To convert from a decimal to a percent:
Multiply the decimal number by 100%. This means you move the decimal point
two places to the right and write the percent sign, %, after the number. Leave
out the decimal point if appropriate.
Example 2
Convert 0.386, 2.45, 0.008 to percent.
Solutions
1.
0.386 = 0.386 100% = 38.6% (move the decimal 2 places to the right)
2.
2.45 = 2.45 100% = 245% (move the decimal 2 places to the right)
3.
0.008 = 0.008 100% = 0.8% (move the decimal 2 places to the right)
Now complete Exercise 1 and check your answers.
MY MATH PATH 6
Module 2 Percent
4
Exercise 1
1.
Convert to a decimal.
a. 37%
______________
b. 2%
______________
c. 47.8%
______________
d. 59.07%
______________
e. 112%
______________
f. 200%
______________
g. 346%
______________
h. 0.13%
______________
i. 5.5%
______________
j. 0.1%
______________
k. 20%
______________
l. 3
1
%
4
______________
1
%
2
______________
n. 24.08%
______________
o. 212%
______________
p. 10%
______________
q. 0.38%
______________
r. 312.6%
______________
s. 87.82%
______________
t. 0.024%
______________
m. 4
2.
Convert to a percent.
a. 0.47
______________
b. 0.83
______________
c. 0.07
______________
d. 0.004
______________
e. 0.0316
______________
f. 0.5
______________
g. 0.6
______________
h. 0.2718
______________
i. 3
______________
j. 1.00
______________
k. 2
______________
l. 3.47
______________
m. 5.009
______________
n. 0.0007
______________
o. 0.38
______________
p. 0.460
______________
MY MATH PATH 6
Module 2 Percent
5
q. 3.1
______________
r. 2.4
______________
s. 7.0060
______________
t. 0.32
______________
3.
A recent survey indicated that 0.25 of all adults chose chicken as their favorite meal.
Find percent notation for 0.25.
4.
92.5% of hockey fans are supporting Vancouver. Find decimal notation for 92.5%.
5.
Lemonade contains 2.5% real lemon juice. Find decimal notation for 2.5%.
6.
On a recent math test 0.18 of the students received marks of 80% or more. What
percent of the students got 80% or more?
7.
A steep hill has a grade of 8%. What is the grade in decimal notation?
8.
The present G.S.T. rate is 7%. Express this rate in decimal notation.
9.
Sales tax in Springville is 0.12 of the purchase price. Express this rate using percent
notation.
Answers are on page 17.
MY MATH PATH 6
Module 2 Percent
6
Unit 2: Converting between fractions and percents
Percent to fractions
To convert from a percent to a fraction:
1.
Write the numeral without a percent sign.
2.
Multiply the numeral by
3.
Reduce the fraction to its lowest terms.
1
.
100
Example 1
Convert 12%,
2
2
%, 16 % and 12.5% to fractions.
5
3
Solutions
1
12
3
=
=
100 100
25
1.
12% = 12
2.
2
2
1
2
1
=
=
%=
5
5 100 500
250
3.
2
50
1
50
1
50
16 % =
%=
=
=
3
3 100
300 6
3
4.
12.5% =
1
12.5
125
=
=
100
1000 8
MY MATH PATH 6
Module 2 Percent
7
Fractions to percent
To convert from a fraction to a percent:
100
%.
1
1.
Multiply the fraction by
2.
Reduce to the lowest terms.
Example 2
Convert
2 3
1
, and 1 to percents.
3 8
2
Solutions
1.
2
2 100
200
2
=
%
% 66 %
3
3 1
3
3
2.
3
3 100
300
1
=
%
% 37 %
8
8 1
8
2
3.
1 100
3 100
300
1
%
%
% 150%
2 1
2 1
2
Now complete Exercise 2 and check your answers.
MY MATH PATH 6
Module 2 Percent
8
Exercise 2
1.
Convert to a fraction.
a. 45%
b. 20%
c. 37%
d. 100%
e. 65%
f. 75%
g. 40%
h. 103%
i. 2%
j. 250%
1
k. 37 %
2
l. 66
m. 12
1
%
2
2
%
3
n. 25%
o. 87.5%
1
p. 33 %
3
2
q. 16 %
3
1
r. 8 %
3
1
s. 83 %
3
t. 6.25%
MY MATH PATH 6
Module 2 Percent
9
2.
Convert to a percent.
a.
1
4
b.
2
5
c.
3
4
d.
7
10
e.
4
5
f.
6
10
g.
1
2
h. 1
i.
8
10
j.
17
100
k.
1
3
l.
1
10
m.
1
6
o.
3
8
p.
1
9
q.
5
8
r.
135
900
s.
20
180
t.
3
11
3
5
n. 2
2
3
Answers are on page 17.
MY MATH PATH 6
Module 2 Percent
10
Unit 3: Adding and subtracting percents
There are many cases where percents must be added or subtracted.
Example 1
The deductions to Frank’s paycheque amounted to 36.2%. What percent of the paycheck did
Frank take home?
The total paycheck must be represented as 100% of the paycheck
100% - 36.2% = 63.8%
Example 2
Sulfuric acid by weight contains 2% hydrogen and 65% oxygen. The rest is sulfur. What
percent of sulfuric acid is sulfur?
The total weight must be represented by 100%
100% - (2% + 65%) = 100% - 67% = 33%
Now complete Exercise 3 and check your answers.
MY MATH PATH 6
Module 2 Percent
11
Exercise 3
1.
Calculate the following.
3
%
4
a. 90% - 35%
b. 15% +
c. 100% - 22.5%
d. 2% +
e. 100% - (12% + 16%)
f. 100% + 7%
1
3
% +9 %
2
4
g. 54% + 19% + 7.9% + 2.8%
2.
3.
4.
1
4
Mary spends 33 % of her time sleeping and 23 % of her time working. What
3
5
percent of her time is spent working and sleeping?
If 11% of Canadians are not employed, what percent are employed?
1
If a shopkeeper discounts (reduces) an item by 33 % , what percent of the old price
3
will be the new selling price?
5.
Sales tax is 6%. What percent of the selling price must you actually pay when the
sales tax is included?
6.
27.2% of carbon dioxide is carbon and the rest is oxygen. What percent of carbon
dioxide is oxygen?
Answers are on page 17.
MY MATH PATH 6
Module 2 Percent
12
Unit 4: Percent increase and decrease
Percent is often used to state increases or decreases. Statements like “the cost of living rose
4%” or “he had to take a 10% decrease to save his job” are common examples.
To find a percentage increase or decrease:
1.
Calculate:
Largest Number - Smallest Number
Original amount
2.
Translate the decimal number obtain in step 1 to percent..
Example 1
The cost of tuition for a full-time college student recently increased from $585.00 to $643.50
per semester. What was the percent increase?
1. The original tuition for the student was $585.00. Using the following formula::
Largest Number - Smallest Number
Original amount
=
2.
643.50 - 585.00
= 0.09914 0.1 (rounding)
585.00
Change from decimal to percent: 0.1 = 0.1x 100% = 10%
Example 2
By changing furnace filters and maintaining the furnace properly, it is estimated that a
monthly heating bill of $84 can be reduced to $73.92. What is the percent of decrease?
1. The original heating bill was $84.00. Using the following formula::
Largest Number - Smallest Number
Original amount
=
2.
84 - 73.92
= 0.12
84
Change from decimal to percent: 0.12 = 0.12 x 100% = 12%
MY MATH PATH 6
Module 2 Percent
13
Now complete Exercise 4 and check your answers.
Exercise 4
1.
Jack’s cheque increased from $800 to $850 a month. What is the percent increase?
2.
Mona will receive $0.65 more per hour next month. What is the percent increase if
she receives $12.05 an hour now? Round your answer to one decimal place.
3.
Sam bought a tape deck for $360 and later sold it for $150. What is the percent
decrease? Round your answer to one decimal place.
4.
Last year Sally made $23,484.32 and paid $3,940.00 in taxes. This year she made
exactly the same, but paid $4,189.50 in taxes.
a. What was the percent increase in her salary?
b. What was the percent increase in her taxes?
Round your answers to one decimal place.
5.
Two years ago, calves were $0.89 per pound. Last year they were $0.75 per pound.
This year they are $0.96 per pound.
a. What is the percent decrease from two years ago to one year ago?
b. What is the percent increase from last year to this year?
Round your answers to one decimal place.
Answers are on page 17.
MY MATH PATH 6
Module 2 Percent
14
Unit 5: Percent applications - sales tax
Percent is widely used in sales tax computations. The provincial sales tax rate in British
Columbia (PST) is currently 7%. This means that the tax is 7% of the purchase price.
Another sales tax, the goods and services tax (GST) is collected for the federal government
and is also 7% of the purchase price. The total price of an article is the price plus PST and
GST.
Example 1
A refrigerator is priced at $699.95 plus PST and GST. How much is the PST, the GST and
the total price?
First, find the PST, 7% of $699.95:
In mathematics, the word “ of “ usually means multiplied. To calculate 7% of $699.95,
change the “of ” to “x”.
7% x $699.95
=
0.07 x $699.95
=
$48.9965
Round to the nearest cent = $49.00
GST is also 7% of $699.95 = $49.00
The total price is $699.95 + $49.00 + $49.00 = $797.95
Now complete Exercise 5 and check your answers.
Exercise 5
1.
The sales tax in a certain city is 8.5%. How much tax is charged on a purchase of
$384.00? What is the total price?
2.
A stove is priced at $449.95. PST is 7% and GST is 7%. Find the amount of the
PST, the GST and the total price of the stove.
Answers are on page 17.
MY MATH PATH 6
Module 2 Percent
15
Practice Test
1.
Convert to decimals:
a. 33%
2.
Convert to fractions:
a. 25%
3.
c. 3
5.
6.
1
b. 83 %
3
Convert to percentage:
a. 0.57
4.
b. 152.3%
1
4
b. 0.0003
d.
1
3
The Smith’s rent went from $375 a month to $400 a month. What is the percent
increase? Round your answer to one decimal place.
Sodium sulfate is 32.4% sodium, 22.5% sulfur and the rest is oxygen. What percent
is oxygen?
In one year, the price of silver fell from $9.20 an ounce to $6.35. What is the percent
decrease? Round your answer to one decimal place.
Answers are on page 18.
MY MATH PATH 6
Module 2 Percent
16
Answers
Exercise 1 - pages 5-6
1. a. 0.37
g. 3.46
m. 0.045
s. 0.8782
b. 0.02
c. 0.478
h. 0.0013 i. 0.055
n. 0.2408 o. 2.12
t. 0.00024
2. a. 47%
b. 83%
c. 7%
g. 60%
h. 27.18% i. 300%
m. 500.9% n. 0.07%
o. 38%
s. 700.6% t. 32%
3.
4.
5.
6.
7.
8.
9.
d. 0.5907
j. 0.001
p. 0.1
e. 1.12
k. 0.2
q. 0.0038
f. 2.00 or 2
l. 0.0325
r. 3.126
d. 0.4%
j. 100%
p. 46%
e. 3.16%
k. 200%
q. 310%
f. 50%
l. 347%
r. 240%
j. 2
1
2
1
p.
3
13
20
3
k.
8
1
q.
6
3
4
2
l.
3
1
r.
12
d. 70%
j. 17%
p. 111/9%
e. 80%
k. 331/3%
q. 62½%
f. 60%
l. 10%
r. 15%
d. 12¼%
e. 72%
f. 107%
25%
0.925
0.025
18%
0.08
0.07
12%
Exercise 2 - page 10
9
20
2
g.
5
1
m.
8
5
s.
6
1. a.
2. a. 25%
g. 50%
m. 162/3%
s. 111/9%
b.
1
5
3
100
1
n.
4
1
t.
16
h. 1
37
100
1
i.
50
7
o.
8
c.
b. 40%
c. 75%
h. 160%
i. 80%
n. 2662/3% o. 37½%
t. 273/11%
d. 1
e.
f.
Exercise 3 - page 12
1. a. 55%
g. 83.7%
2. 572/15%
3. 89%
4. 662/3%
5. 106%
6. 72.8%
b/ 15¾%
MY MATH PATH 6
c. 77.5%
Module 2 Percent
17
Exercise 4 - pages 14
1.
2.
3.
4.
5.
6.25%
5.4%
58.3%
a. 0%
b. 6.3%
a. 15.7% b. 28%
Exercise 5 - page 15
1.
2.
3.
4.
5.
6.
7.
8.
$32.64 tax, $416.64 total
$31.50 PST, $31.50 GST, $512.95 total
$433.80
$14,336 Saskatchewan, $14,592 BC
8%
4%
$90
$1,171.43
Practice Test - pages 16
1. a. 0.33 b. 1.523
2. a. ¼
b. 5/6
3. a. 57% b. 0.03%
7. 6.6% or 6.7%
8. 45.1%
10. 31.0%
MY MATH PATH 6
c. 325%
d. 331/3% or 33.3%
Module 2 Percent
18
Moodle Courses and Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
MY MATH PATH 6
Moodle Course Developers – Capilano University
Students in the College and University Preparation
Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
Module 2 Percent
19
MY MATH PATH 6 – MODULE 3
POWERS AND ROOTS
Contents
Learning outcomes .....................................................................2
Glossary .....................................................................................3
Introduction ................................................................................4
Unit 1: Powers - repeated multiplication ..................................5
Exercise 1 .........................................................................8
Unit : Square roots ................................................................10
Exercise .......................................................................11
Practice Test .............................................................................13
Answers....................................................................................14
MY MATH PATH 6
Module 3 –Powers and Roots
1
Learning outcomes
Exponents and roots are used throughout algebra, and in other sciences as well. A scientific
calculator can be used to find any required root or power of a number.
When you have completed this Module, you should be able to:
understand and use roots and powers
find a root or a power using a scientific calculator
MY MATH PATH 6
Module 3 –Powers and Roots
2
Glossary
Base
The number that is multiplied by itself a certain number of times, depending on the exponent.
Cube
To cube a number means to raise it to the third power.
Exponent
The small number placed above and to the right of a base number. It indicates how many
times that base is to be multiplied by itself.
Power
A base number raised to a given exponent.
Radical
An expression involving the symbol
.
Radicand
In the expression
5 , 5 is called the radicand.
Root
A number that, multiplied by itself a given number of times, equals a radicand.
Square
To square a number means to raise it to the second power.
Square root
A radical with an index of two (often omitted).
MY MATH PATH 6
Module 3 –Powers and Roots
3
Introduction
There is a story about the mathematician who invented the game of chess. He presented this
new game to the King, who was very impressed. The King announced that the
mathematician could name his reward for inventing such a wonderful game. The
mathematician thought for a moment, and then asked that he be given some rice for his
family - but that the amount of rice be counted as follows:
“On the first square of the chess board place one grain of rice, on the second square 2, on the
3rd square 4, on the 4th square 8, and so on, doubling the number each time.”
Now, a chessboard has 64 squares, so the number of rice grains on the last square of the
chessboard will be 263 or about 9,223,372,000,000,000,000.
This amount of rice would be greater than all of the rice ever harvested in the history of the
world. This module deals with ways to handle such giant numbers, plus other numbers that
use exponents.
MY MATH PATH 6
Module 3 –Powers and Roots
4
Unit 1: Powers - repeated multiplication
As you may recall, multiplying is a way of indicating that we should add the same number so
many times. For example,
53=5+5+5
The power operation is a way of indicating that we should multiply the same number so many
times. For example,
53 = 5 5 5
The power operation is indicated by placing a small number, called an exponent, to the top
right of some number, or base, that we then multiply so many times together.
53 is called a power.
5 is called the base.
3
is called the exponent.
The power, 53, is read “five to the third” or more commonly, “five cubed”. Powers with
exponents of 2 are said to be “squared”, while those with exponents of 3 are said to be
“cubed”. Below are some examples of how we read powers.
72 is read “seven squared”
73 is read “seven cubed”
94 is read “nine to the fourth”
128 is read “twelve to the eighth”
Notice that the exponent is not really a number - it is really an instruction: it says “multiply
the number below me by itself this many times”. So 94 = 6561, not 36.
We can use exponents to find powers of integers, decimals and fractions:
(-3)5 = (-3)(-3)(-3)(-3)(-3) = -243
(1.25)3 = (1.25)(1.25)(1.25) = 1.953125
256
28 2 2 2 2 2 2 2 2
= =
3
3 3 3 3 3 3 3 3 6561
You probably realize now why exponents are used - they save a lot of time writing down
repeated multiplication.
Scientific calculators have a “raise to the power” key, usually labelled yx (sometimes xy or ax
is used). To use this function, press the yx key in between the base number and the exponent,
then press = .
MY MATH PATH 6
Module 3 –Powers and Roots
5
Example 1
To find
35 = ? press:
yx
3
5
=
Did you get 243?
Example 2
To find
(1.25)3 = ? press: 1.25
yx
3
=
yx
10
=
Did you get 1.953125?
Example 3
To find
(2.57)10 = ? press: 2.57
Did you get 12569.88294?
Take a look at the following patterns:
24 = 2 2 2 2
23 = 2 2 2
22 = 2 2
21
20
104 = 10 10 10 10
103 = 10 10 10
102 = 10 10
101
100
= 16
=8
=4
=2
=1
= 10000
= 1000
= 100
= 10
=1
Notice that each line is equal to one factor (2 or 10) divided into the line above it. It seems
reasonable the 21 = 2 or 101 = 10. After all, the exponent gives the number of factors that are
multiplied together. But why should 20 = 1 or 100 = 1 ? A zero exponent does look odd, but
it has to follow the same pattern:
23 2 = 22 = 4
103 10 = 102 = 100
21 2 = 20 = 1
101 10 = 100 = 1
Note that your calculator can use zero as an exponent, except in the case 00, which is
undefined. Otherwise, any base number raised to the 0 power equals 1.
Finally, here is a shortcut you can use on your calculator: The most common exponent used
in algebra is 2. Your calculator will have a key labelled x2. You can use this key instead of
the yx key, and save yourself some time:
13
x2
or
13
x2
=
Did you get 169?
MY MATH PATH 6
Module 3 –Powers and Roots
6
Fractions can also be squared as follows:
2
22
2
2
3
3
To square the 2 and 3 separately:
x2 is 4 and
2
3
x2 is 9 and rewrite as
2
4
2
9
3
Now complete Exercise 1 and check your answers.
MY MATH PATH 6
Module 3 –Powers and Roots
7
Exercise 1
1.
2.
How would you read the following?
a. 32
_______________________________________________
b. 103
_______________________________________________
c. x3
_______________________________________________
1
d. 4
2
_______________________________________________
e. 66
_______________________________________________
Write the following as powers.
a. 2 2 2
_______________________________________________
b. 8 8 8 8 8
_________________________________________
1 1
c.
4 4
_________________________________________
d. (9.7)(9.7)(9.7)(9.7) _________________________________________
e. 5x5x5x5x5x5x5x5x5
___________________________________
f. 26 _____________________________________________________
3.
Evaluate the following:
a. 132 = _____________
b. 142 = _____________
c. 152 = _____________
d. 202 = _____________
e. 252 = _____________
MY MATH PATH 6
Module 3 –Powers and Roots
8
1
f. 2
2
= _____________
1
g. 3
2
= _____________
1
h. 4
2
= _____________
i. (0.5)3
= _____________
2
j. 3
3
= _____________
k. (5.3)1
= _____________
= _____________
5
m. 2
8
= _____________
= _____________
o. (0.1)5
= _____________
l.
90
n. (1.2)2
4.
The following are called “powers of ten”. Evaluate them.
a. 100 = _____________
b. 101 = _____________
c. 102 = _____________
d. 103 = _____________
e. 104 = _____________
f. 105 = _____________
g. 106 = _____________
h. 107 = _____________
i. Get the idea? = _____________________________________________
Answers are on page 13.
MY MATH PATH 6
Module 3 –Powers and Roots
9
Section 2: Square roots
A root is the “reverse” of a power. Since 52 = 25, we say that 25 is the square of 5. We could
reverse this statement, and say that 5 is the square root of 25. In symbols, this would be
written:
25 = 5
The symbol
means “find the number, that when squared, equals the number inside”. We
call the symbol
expression
a radical and the number inside it is the radicand. The radicand in the
25 is 25. The expression
25 reads “the square root of 25.”
Take a look at the following:
9 = 3 because 3 3 = 9
49 = 7 because 7 7 = 49
100 = 10 because 10 10 = 100
0.25 = 0.5 because 0.5 0.5 = 0.25
1
1
1
1
1
= because
=
16
4
4
4 16
What about numbers that do not have whole number square roots? For example, what is the
square root of 30?
Look for the
button on your calculator. Use this button to find the above square roots.
Did you get the right answers? Try the following, using your
2 =
Did you get 1.414213562?
195 =
Did you get 13.96424004?
0.05 =
Did you get 0.223606797?
2
=
3
Did you get 0.81649658?
MY MATH PATH 6
Module 3 –Powers and Roots
button.
10
A = b because b b = A.
Now complete Exercise 2 and check your answers.
Exercise 2
1.
a.
9 =
because
=9
b.
81 =
because
= 81
c.
25 =
because
= 25
d.
100 =
because
= 100
e.
4 =
because
=4
f.
0 =
because
=0
g.
1 =
because
=1
h.
4900 =
because
= 4900
i.
1600 =
because
= 1600
j.
1
=
4
because
=
MY MATH PATH 6
Module 3 –Powers and Roots
1
4
11
k.
1
=
25
because
=
1
25
l.
4
=
9
because
=
4
9
16
25
m.
16
=
25
because
=
n.
1.21 =
because
= 1.21
o.
0.64 =
because
= 0.64
p.
0.01 =
because
= 0.01
Answers are on page13.
MY MATH PATH 6
Module 3 –Powers and Roots
12
Practice Test
1.
Write the following as powers.
a. 3 3 3
1
b.
9
_______________________________________________
1
9
_________________________________________
c. (0.7)(0.7)(0.7)(0.7) _________________________________________
d. 14 _____________________________________________________
2.
Evaluate the following:
a. 130 = _____________
b. 92 = _____________
c. 152 = _____________
d. 1252= _____________
e. 12 = _____________
1
f. 5= _____________
2
3.
Evaluate the following:
1.
a.
4 =
b.
64 =
c.
0 =
d.
1 =
e.
7921 =
f.
1
=
16
g. (2.1)3= _____________
Answers are on page 13.
MY MATH PATH 6
Module 3 –Powers and Roots
13
Answers
Exercise 1 - pages 71.
a. three squared
b. ten cubed
d. one half to the fourth
3
2.
a. 2
b. 8
3.
a. 169
b. 196
1
8
25
m.
64
a. 1
g. 1000000
g.
4.
h.
1
c.
4
5
1
16
n. 1.44
d. 9.74
e. 510
f. 261
c. 225
d. 400
e. 625
f.
i. 0.125
j.
k. 5.3
l. 1
b. 10
c. 100
d. 1000
e. 10000
f. 100000
h. 10000000 i. number of zeros = exponent for powers of ten
a. 3
b. 9
c. 5
g. 1
h. 70
i. 40
n. 1.1
o. 0.8
4
m.
5
8
27
1
4
o. 0.00001
Exercise 2 - pages 11
1.
2
c. x cubed
e. six to the sixth
d. 10
1
j.
2
p. 0.1
e. 2
1
k.
5
f. 0
2
l.
3
Practice Test - pages 123
1. a. 3
b.
2. a. 1
b.
e. 1
f.
3. a. 2
b.
e. 89
MY MATH PATH 6
f.
1
9
81
1
32
8
1
4
2
c. (0.7)4
d. 141
c. 225
d. 15625
g. 9.261
c. 0
Module 3 –Powers and Roots
d. 1
14
Moodle Courses and Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
MY MATH PATH 6
Moodle Course Developers – Capilano University
Students in the College and University Preparation
Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
Module 3 –Powers and Roots
15
MY MATH PATH 6 – MODULE 4
STATISTICS
Contents
Learning outcomes .....................................................................2
Glossary .....................................................................................3
Introduction ................................................................................4
Unit 1: Interpreting graphs ........................................................5
Exercise 1 .........................................................................8
Unit 2: Mean, median, mode and range ..................................15
Exercise 2 .......................................................................18
Practice Test .............................................................................21
Answers....................................................................................25
MY MATH PATH 6
Module 4 Statistics
1
Learning outcomes
Statistics is the branch of mathematics that helps you collect, organize and find meaning in
numerical facts. The word statistics also refers to the numerical facts themselves.
When you have completed this Module, you should be able to:
calculate median, mean, mode and range
interpret bar, line, and circle graphs
MY MATH PATH 6
Module 4 Statistics
2
Glossary
Bar graph
A type of graph that uses side-by-side horizontal bars or vertical bars of different lengths to
compare different items of information.
Circle graph
A type of graph that is circular in shape. The circle graph compares different information by
assigning “pie Units” of different sizes to the various categories of information.
Data
Factual information you collect to solve a problem.
Graph
A diagram that “pictures” data or information in a way that makes it easier to understand. Common
graphs are bar graphs, circle graphs and line graphs.
Interpret
The way you explain or arrive at a conclusion based on a study of data.
Line graph
A series of points, plotted between the horizontal and vertical axes of a graph, that form a line
when connected to one another. Each point on the line represents specific values of the quantities
labelled along the horizontal and vertical axes of the graph.
Mean
The sum of all the values in a set divided by the number of values in the set.
Median
The value in the middle of a data set when the data are arranged in numerical order. There are an
equal number of data values greater than the median value and an equal number of data values less
than the median value.
Mode
The number that appears most often in a set of data.
Range
The difference in the maximum and minimum values for a set of data.
MY MATH PATH 6
Module 4 Statistics
3
Introduction
Statistics is the branch of mathematics that helps you collect, organize and find meaning in
numerical facts or data. Data can be organized into a table and then used to construct a
graph. Figure 1 shows data on level of education and employment in 1996.
Education
Percentage of employment
Grade 8 or less
20%
Some secondary
40%
Graduated from high school
62%
Some college/university
64%
Post-secondary diploma
69%
University degree
77%
Figure 1
This type of statistical information frequently appears in newspapers and magazines. Note
how easy it is to read and interpret the data. For example, what conclusion or conclusions
would you draw about higher education and employment?
MY MATH PATH 6
Module 4 Statistics
4
Unit 1: Interpreting graphs
Graphs are convenient devices for showing data or illustrating information. Since they are
drawn, rather than written, a graph makes it possible to get a quick impression of a great deal
of data and to easily make comparisons and draw conclusions. We can see various kinds of
graphs in newspapers and magazines as well as on the television. Graphs are particularly
easy to produce using a computer. There are three types of common graphs:
bar
line
circle
Although these graphs all appear different, they have properties in common that you must
know if you are to correctly construct your own graphs. All graphs have to be titled and
labelled so that everyone looking at or analyzing the graph will get the same information. As
you go through the graphing Units, pay particular attention to how the graphs are all titled
and labelled.
How to read any graph
You can get information from a graph quickly if you follow these steps:
1.
2.
3.
4.
Read the title of the graph.
Read the labels and range of numbers (scale) along the sides.
Determine what units the graph uses.
Look for groups, patterns and differences.
Bar graphs
For example, the graph in Figure 2 can
be easily used to answer the question:
“Which kinds of television programs
were chosen as a favorite by 50 or
more college students?”
Detective and western shows were
chosen as favorites by 50 or more
students.
MY MATH PATH 6
60
50
Number of students
The bar graph is used to show
comparisons and trends. Bar graphs
get their name from the thick bars with
which they are drawn. The bars may
be vertical or horizontal. A small
space usually separates each bar. The
bars are often different colours.
Favourite Television Shows
40
30
20
10
0
Detective
Comedy
Variety
Western
Type of TV show
Figure 2 Bar graph
Module 4 Statistics
5
Line graphs
Line graphs plotted from sets of data are
good for showing trends and developments
over a period of time.
A line graph can be drawn with either
straight or curved lines that extend across
the graph in a horizontal direction. The
graph in Figure 3 shows how the number of
farms has dropped steadily over a thirty-year
period up to 1970.
Often it is necessary to find a particular
quantity from a line graph. For example, to
Figure 3 Line graph
answer the question “How many millions of
farms were there in 1960?” locate 1960 on the horizontal scale. Move up to the point on the
graph. Then move left to the number on the vertical scale. The vertical scale shows that
there were 4 million farms in 1960.
Circle graphs
A circle graph shows an entire quantity divided into various parts. Each part of the circle is
called a Unit (or sector) and has its own name and value. Circle graphs are also known as
“pie” graphs because they can look like a pie cut into pieces. In most cases, values on circle
graphs consist of either parts of a dollar or a fraction or a percent of a whole.
An example of a circle graph showing the expenses of a student going to a community
college is shown in Figure 4.
Figure 4 Circle graph
MY MATH PATH 6
Module 4 Statistics
6
This circle graph clearly shows which college expense is the greatest and which is the least.
The Unit of the circle graph showing tuition is largest; so tuition is the greatest expense.
The circle graph can be used to calculate certain different data. For example, you can use the
circle graph in the example above to find how much money would be spent on tuition if the
total college expenses were $2400 for a year.
Find 23% of $2400
0.23 2400 = 552
Write 23% as a decimal and multiply.
Tuition would be $552 per year.
Now complete Exercise 1 and check your answers.
MY MATH PATH 6
Module 4 Statistics
7
Exercise 1
Use this graph to answer questions 1 to 3.
1.
Which material makes up the greatest part of the human body?
2.
Which materials make up 20% or less of the human body?
3.
If a student has a mass of 64 kg, about how much of the body mass is:
a. fat
b. bone
c. muscle
d. other
MY MATH PATH 6
Module 4 Statistics
8
Use the following graph to answer questions 4 to 13.
Suppose you looked back at your old bills and graphed the amount of money paid each bimonthly billing for electricity in 1995 and 1996.
Which billings in 1995 were:
4.
more than $150?
5.
less than $100?
6.
$150 or less?
Which billings in 1996 were:
7.
less than $100?
8.
$150 or more?
9.
$165 or less?
Find the total amount of the hydro bill for:
10.
1995
11.
1996
Hydro users can choose to make equal monthly payments. Each payment is the average of
last year’s bills. Find the calculated equal monthly payment for each of these years. Round
answers to the nearest cent.
MY MATH PATH 6
Module 4 Statistics
9
12.
1996
13.
1997
Use the following graph for questions 14 to 18.
14.
What are the units of measurement for altitude and for temperature in the graph?
15.
How much has the temperature decreased from sea level to an altitude of 3,000 feet?
16.
Think about extending the line to estimate the temperature at 10,000 feet and at
15,000. How reliable do you think these estimates are? Why?
17.
Think about extending the line to estimate the temperature at 500 feet below sea level.
How reliable do you think this estimate would be? Why?
18.
Why would early aviators have been interested in the graph?
MY MATH PATH 6
Module 4 Statistics
10
Refer to the graph to answer questions 19 to 22.
You work as a quality control supervisor at the Compu-Rite Parts factory. A technician
prepares a report on the rejected data boards with a graph, as shown.
19.
What dates does the defect report cover? How many days of production data are
shown?
20.
What is the highest level of defects reported on the graph?
21.
What is the lowest level of defects reported on the graph?
22.
On what day did the highest level of defects occur?
MY MATH PATH 6
Module 4 Statistics
11
Based on the bar graph, answer questions 23 to 27.
As the purchasing agent for the electrical department, you are responsible for keeping a
supply of dry-cell batteries for hand-held equipment used by the technicians. A trade
magazine shows that some types of batteries perform better than others. The report displays a
bar graph, as shown here.
23.
Which type of battery lasts the longest under either high-rate or low-rate use?
24.
The department is using heavy-duty batteries now. How do they compare to the other
types of batteries available?
25.
The department needs batteries in flashlights to last through a minimum 12-hour
shift of high-rate use. Which battery types would you select to meet this
requirement?
26.
Under conditions requiring only occasional, low-rate use — such as portable radio
gear — batteries should last at least 48 hours. Which battery types would you select
to meet this requirement?
27.
Cost is an important factor. What does the graph say about the cost of each type of
battery?
MY MATH PATH 6
Module 4 Statistics
12
Use the circle graph to answer questions 28 to 30.
The circle graph shown here is about types of treatment at the Cedar Hill Walk-in Clinic.
28.
What kind activity was provided most often at the Clinic during the year?
29.
Compare the number of minor surgery treatments to the number of routine
examinations. There are about ____________ times as many routine examinations as
there are minor surgery treatments.
30.
Rank the types of treatment given at the clinic this year, starting with the most
frequent, down to the least frequent.
The bar graph below is about electronic appliances. Study the bar graph and then answer
questions 31 to 35.
MY MATH PATH 6
Module 4 Statistics
13
31.
Should automobiles be included in the list of appliances? Why or why not?
32.
How many kinds of appliances are shown?
33.
What appliances are used in over half of the households?
34.
What appliances are used in less than half of the households?
35.
What other information do you need to find how many households have home
computers?
The following bar graph shows the average braking distance required for a car with antilock
brakes travelling at different speeds on dry pavement. The braking distance is measured from
the time the brakes are applied until the car comes to a complete stop. Refer to this graph to
answer questions 36 and 37.
36.
What is the unit of measurement used to indicate the braking distance?
37.
What is the braking distance shown for a car traveling at a speed of 80 kilometres per
hour?
Answers are on pages 25-26.
MY MATH PATH 6
Module 4 Statistics
14
Unit 2: Mean, median, mode and range
The mean, median and mode are three different measures of the “middle” of a set of data.
Finding the mean
The average of a set of data is called the mean.
Suppose six people are weighed and their mass in kilograms each recorded as follows and
you want to find the mean (average) mass in kilograms for this set of six people.
40
52
60
96
39
83
To find the mean, first add the items.
40 + 52 + 60 + 96 + 39 + 83 = 370
370 is the sum of the items. Then divide the sum by the number of items. It is convenient to
round up the answer to the nearest tenth.
370 6 = 61.7
The mean mass is 61.7 kg.
the mean =
sum of the items
number of items
Finding the median and the mode
The mean gives you some information about a set of numbers or data. The median and the
mode are two other different measures of a set of data.
Suppose 15 families have the following numbers of children:
1, 3, 5, 3, 7, 1, 0, 1, 2, 8, 3, 2, 0, 1, 2
The mean of these values is 2.6, yet no family has 2.6 children and this is not a meaningful
value in this situation. Instead, re-arrange the data in order. The median is the middle
number. It has as many numbers above as it has below.
The median family has 2 children.
MY MATH PATH 6
Module 4 Statistics
15
In the following set of data, with an even number of quantities, there are two middle
numbers. The median is halfway between these two numbers.
90
85
84
56
84.5 is the median
The median is a quantity where half the numbers in the set of data are above it
and half are below. Remenber that the data must be arranged in order first.
The mode is the most common number that appears in a data set. Using the same 15
families, select the number that appears most often.
The mode is 1 child per family.
The mode is the number that appears most often in a set of data.
Note that some data sets could have more than one mode, while others may not have a mode.
For example, if in the 15 families there was one less family with one child, there would
actually be three modes; 1, 2, and 3. If all the families had different numbers of children
there would be no mode.
The mean, median and mode in a set of data may be the same number or they may all be
different.
MY MATH PATH 6
Module 4 Statistics
16
Finding the range
To find the range in a set of numbers, find the difference between the largest and smallest
number in the set. The difference is the range.
To determine range consider the above data:
largest number
smallest number
range
=8
=0
=8-0
=8
The range is the difference between the largest and smallest number in a data
set.
Now complete Exercise 2 and check your answers.
MY MATH PATH 6
Module 4 Statistics
17
Exercise 2
Find the mean, median, mode and range for the following sets of data:
1.
Math students test scores: 73, 100, 97, 85, 59, 72, 85, 91, 96
mean =
median =
2.
range =
Temperatures for a one-week period: 0, -2, -3, -2, 0, 1, 2
mean =
median =
3.
mode =
mode =
range =
Estimates from various repair shops: $415, $375, $400, $500, $325, $425
mean =
median =
MY MATH PATH 6
mode =
range =
Module 4 Statistics
18
4.
Grade point averages for 10 students: 2.0, 1.9, 3.2, 3.5, 2.8, 2.5, 2.0, 3.4, 2.4, 3.4
mean =
median =
5.
range =
The number of hours of television viewed per day for one person for 9 days: 2, 4, 0,
1, 2, 0, 3, 2, 4
mean =
median =
6.
mode =
mode =
range =
Salaries at Gigantic Corporation: $120,000, $90,000, $40,000, $30,000, $30,000,
$30,000, $30,000, $25,000
mean =
median =
MY MATH PATH 6
mode =
range =
Module 4 Statistics
19
7.
Tips over a one-week period: $7.50, $10, $9.75, $12.00, $9.50, $20.25, $8.00
mean =
median =
8.
mode =
range =
Shoe sizes for a group of ten women: 7 1/2, 5, 5 1/2, 7, 8, 9, 6, 7, 8 1/2, 6 1/2
mean =
median =
mode =
range =
Answers are on page 26.
Now complete the Practice Test and check your answers.
MY MATH PATH 6
Module 4 Statistics
20
Practice Test
1.
During a semester, four students received these grades on their math tests:
Manuel
70
70
60
65
85
Abagail
56
58
81
92
58
Terry
48
35
60
58
60
Han
74
48
65
74
95
a.
Complete the following table to summarize the students’ marks for the
semester (the students’ names are identified by their first initials):
Mean
Median
Mode
Range
M
A
T
H
b.
Which of these students would you rank as the best of the four and why?
c.
Which of these students would you say is the most consistent? Give a reason
for your answer.
MY MATH PATH 6
Module 4 Statistics
21
2.
Use this bar graph to answer the following questions:
a.
What reason was most common for dropping out?
b.
What percentage dropped out because of grades?
c.
Why would the total percentage of the four bars in this graph be more than
100%?
MY MATH PATH 6
Module 4 Statistics
22
3.
Use this line graph to answer the following questions:
a.
In which month did Matt’s team win the fewest games?
b.
In October, how many games did Matt’s team win?
MY MATH PATH 6
Module 4 Statistics
23
4.
Use the graph below to answer the following questions:
a.
What was Jim’s average driving speed for the round trip?
Hint: speed = distance time
b.
At what time did Jim leave Qualicum?
MY MATH PATH 6
Module 4 Statistics
24
Answers
Exercise 1 - pages 8-14
1. muscle
2. fat, bone and other
3. a. 12.8 kg
b. 11.52 kg
c. 32 kg
d. 7.68 kg
4. February
5. none
6. all but February
7. August, October
8. February, April, December
9. April, June, August, October
10. $770
11. $825
12. $64.17
13. $68.75
14. The units of measurement used in the graph are feet for altitude and degrees Fahrenheit
for temperature.
15. From sea level to an altitude of 3 000 feet, there is a 13 degree drop in temperature.
16. The estimate for 10 000 feet and 15 000 feet would not be very accurate because you
cannot be sure that your curve is accurately drawn.
17. Hint: Are you 500 ft. under water or in the air over Death Valley, California? If the
latter, the graph can be used to estimate air temperature.
18. Early aviators could have used the information in the graph to determine how to dress for
the temperature changes and to determine the best altitudes at which to fly to avoid
problems associated with temperature.
19. The defect report covers the dates Oct 2 through Oct 13. Only ten days of production
data are shown.
20. The highest level of defects shown is 3.75%.
21. The lowest level of defects is 1.25%.
22. The highest level of defects occurred on Oct 2.
23. The alkaline battery lasts longer under either high- or low-rate use.
24. Heavy-duty batteries are better than general purpose or rechargeable batteries, but not as
long-lasting as alkaline batteries.
25. Only the alkaline battery seems to last through more than 12 hours of high-rate use. The
heavy-duty batteries come close.
26. Either the alkaline or the heavy-duty batteries last at least 48 hours when used in low-rate
applications.
27. The graph tells you nothing about the cost of these types of batteries.
28. Routine examinations were provided most often.
29. There are about four times as many routine examinations as minor surgery.
30. Ranking — from most frequent to least frequent — is: routine examinations,
prescription renewals, common cold symptoms, minor surgery, others.
MY MATH PATH 6
Module 4 Statistics
25
31. No, this is a graph of “electronic appliances”. Automobiles do not fit in this category.
32. Eight kinds of appliances are shown.
33. The last three in the graph are used in over half (more than 50%): stereo systems, TVs
and radios.
34. The first five appliances are used in less than half of the households.
35. You need to know how many total households there are in order to apply the percentage
and find the number that have home computers.
36. The unit of measure used to indicate the braking distance is metres.
37. The braking distance for a car traveling at 80 kph is about 34 m.
Exercise 2 - pages 18-20
1. mean = 84.2
median = 85
2. mean = -0.6°
median = 0°
3. mean = $406.67
median = $407.50
4. mean =2.71
median = 2.65
5. mean = 2
median = 2
6. mean = $49,375
median = $30,000
7. mean = $11
median = $9.75
8. mean = 7
median = 7
mode = 85
range = 41
mode = -2 and 0
range = 5°
mode = none
range = $175
mode = 3.4 and 2.0
range = 1.6
mode = 2
range = 4
mode = $30,000
range = $95,000
mode = none
range = $12.75
mode = 7
range = 4
Practice Test - pages 21-25
1. a.
Mean
Median
Mode
Range
M
70
70
70
25
A
69
58
58
36
T
52.2
58
60
25
H
71.2
74
74
47
b. Student H
c. Manuel and Terry - smallest ranges
2. a. boredom
b. 30%
c. some people must have given more than one reason
3. a. November
4. a. 40 km/hr
MY MATH PATH 6
b. 6
b. 4:
Module 4 Statistics
26
Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
MY MATH PATH 6
Moodle Course Developers – Capilano University
Students in the College and University Preparation
Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
Module 4 Statistics
27
Moodle Courses and Downloading Information
http://www.mymathpath.com
MyMathPath.com Resources:
Copyright © 2011 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Permission granted to make copies for educational purposes.
Based on the Resources:
Adult Literacy Fundamental Mathematics, Instructor’s Manual and Test-Bank
Copyright © 2010 Province of British Columbia
Ministry of Advanced Education and Labour Market Development
Acknowledgments:
Textbook: Sylvie Trudel
Moodle Course Developers - Faculty:
Don Bentley
Robert Kim
Sylvie Trudel
Moodle Course Developers – Capilano University Students
in the College and University Preparation Program:
Crystalina Alcazar
Ivan Arzave
Jasmine Blanchard
Bill Brown
Derek Brownbridge
Michael Drummond
Kristofer Dukic
Kameron Fakhari
Mackenzie Grant
David Hart
Abbid Jaffer
Alexander Kay
Michael Kulch
Nicholas Schopp
Bobbi-dee Stillas
Ksenia Teder
Daniel Thomas
Michael Villareal
Ashley Wong
