Chapter 6
Pre-Algebra
T McDowell
Proportions
11/09
Proportions - If two ratios are equal, they form a
proportion. Proportions can be used in
geometry when working with similar
figures.
1
2
Similar
=
4
8
1:3 = 3:9
- Similar describes things which have the
same shape but are not the same size.
Cross
Multiplying
If a/b = c/d then ad = bc
a
c
b = d
ad = bc
Examples
1. 2/3 = 4/6
2. 10/x = 6/3
3. 5/6 = x/72
Ratio
The ratio of the smaller figure to the larger
figure is 1:2 (said “one to two”). This can also
be written as a fraction of ½.
Proportion A proportion
can be made
relating the
height and the
4 ft
width of the
smaller figure to
the larger
figure: 4 ft 8 ft
=
2 ft
4 ft
8 ft
2 ft
4 ft
Solving
First, designate the
Proportion unknown side as x. Then,
Problems set up an equation using
proportions. What does
the numerator represent? height
What does the
denominator represent? width
4 ft
8 ft
=
2 ft
x ft
Then solve for
x by cross
multiplying:
4x = 16
X=4
8 feet
4 feet
2 feet
? feet
Similar Shapes
11/10
Similar shapes are very important
because if we know the dimensions
of one shape and one of the
dimensions of another shape similar
to it, we can figure out the unknown
dimensions.
You Try These two stick figures are similar.
1. Write a proportion
relating the similar
shapes.
2. Find the missing
width.
8 feet
12 feet
4 feet
x feet
You Try These two trapezoids are similar.
1. Write a proportion
relating the similar
shapes.
15
a
2. Find the missing
sides.
10
40
24
x
You Try The scale of a map is 1 inch : 10
miles. Find the actual distance
given the distance on the map.
1. 4 inches
2. 10 inches
3. 1 foot
4. 5.5 inches
5. 6.75 inches
Leonardo
da Vinci
1452 - 1519
Write a
ratio that
represents
each
statement.
The average adult human figure is about 7
to 7.5 heads tall.
7 head heights
1 body height
The arms' wingspan (measured from the tips
of the middle fingers) is about equal to the
body height.
1 wingspan
1 body height
The length of the foot is about equal to the
length of the forearm.
1 foot length
1 forearm length
da Vinci
Proportions
Activity
Measure in
inches
Head Height
Estimated total height
Wingspan
Estimated total height
Actual height
Foot length
Estimated
forearm length
Actual forearm length
•The eyes are at the mid-height of the head.
•The head also can be divided into thirds
•top of the head to the bottom of the forehead
•bottom of the forehead to bottom of the nose
Use these
•bottom of nose to the bottom of the chin.
proportions
to draw a •Width of head is between four and five eyes wide.
head.
•Height of the face is about equal to length of
hand.
•Eyes are apart by a distance of one eye width.
•Bottom of the nose to the corner of the eye is
equal to the height of the ear.
•Width of base of nose is equal to width of the eye.
•The width of the mouth is equal to the distance
between pupils, or the width of two eyes.
Draw like
de Vinci
Ratios, Decimals, and Percents 11/16
Percent
A ratio that compares a number to
100
Examples 54/100 = 54%
36/100 = 36%
4/25 = 16/100 = 16%
Percents Percents can also be converted into
as
fractions.
Fractions
Place the percent over 100
Reduce the fraction into simplest form.
Example
88%
88 
100 4
4
22
25
You try
Write each fraction as a percent
1. 67/100
2. 7 ¾
3. 32/50
Write each percent as a fraction
1. 92%
2. 48%
3. 326%
Percents
as
decimals
Since percents can be written as
fractions, they can also be converted
to decimals
The fastest way to convert a percent to
a decimal is to move the decimal 2
hops left.
Examples 37%
37.%
0.37
Decimals Decimals can also be converted to
percents
as
Percents
The fastest way to convert a decimal
to a percent is to move the decimal 2
hops right.
Examples 3.45
345%
You try
Convert each percent into a decimal
1. 25%
2. 457%
3. 0.4%
Convert each decimal into a percent
1. 0.89
2. 0.056
3. 9.97
You try
Workbook
P 101
# all
•Turn in homework
•Sharpen pencil
•Sit down
•Get ready for notes
Proportions and Percents 11/17
Proportions One way to solve problems
involving percents is to set up a
proportion.
What is 45% of 60?
We know 45% is 45/100, but we don’t
know what part of 60 we need so that
is x/60
45 = x
100 60
Solve the proportion by cross
multiplying
You try
Write a proportion and solve.
1. 23% of 158
2. 15% of 24
3. 345% of 106
Finding the What percent of 86 is 4?
percent
You know you are looking for a
percent or x/100
x = 4
100 86
Solve the proportion by cross
multiplying
You try
Write a proportion and solve.
1. What percent is 56 of 109?
2. What percent is 3 of 9?
3. What percent is 150 of 80?
Finding the 34 is 67% of what number
whole
amount You have the percent so write that as a
ratio: 67/100
34 is the numerator of the other ratio—
we don’t know the denominator: 34/x
67 = 34
100 x
Solve the proportion by cross
multiplying
You try
Write a proportion and solve.
1. 12 is 56% of what number?
2. 54 is 120% of what number?
3. 21 is 5% of what number?
Know
what
you are
looking
for
A tile floor has 90 blue tiles, which is 15% of
all the tiles in the floor. How many tiles are in
the floor in all?
You have the percent so write that as a
ratio: 15/100
90 is only part of the whole floor so it
is the numerator of the other ratio—we
don’t know the denominator: 90/x
15 = 90
100 x
Solve the proportion by cross
multiplying
You try
Workbook
P 103
# all
•Turn in homework
•Get your workbook
•Sharpen pencil
•Sit down
•Get ready for notes
Percents and Equations 11/18
Math Words Of means to multiply
Is means equal sign
Finding
the part
What is 35% of 90?
x = 35% x 90
x = 0.35 x 90
x = 31.5
You try
Write an equation to solve
1. What is 14% of 65?
2. What is 135% of 15?
3. What is 82% of 110?
Finding
the Percent
Of means to multiply
Is means equal sign
20 is what percent of 120?
20 = x • 120
 120
 120
0.167 = x
16.7% = x
You try
Write a proportion and solve.
1. 12 is what percent of 90?
2. 90 is what percent of 82?
3. 34 is what percent of 150?
Finding
The whole
Of means to multiply
Is means equal sign
15 is 45% of what number?
15 = 45% • x
15 = 0.45 • x
 0.45  0.45
33.3 = x
You try
Write a proportion and solve.
1. 24 is 42% of what number?
2. 145 is 110% of what number?
3. 5 is 30% of what number?
You try
Workbook
P 105
# all
Binder Check
1. What was the topic for the notes given
on 11/18?
2. What was the answer to number 55
from the homework assigned 11/16, p
313, # 55-70
3. Write the calculator policy from the
Classroom Guidelines and Procedures
handout.
Writing Proportions11/30
Similar
Shapes
34
10
26
x
Write a proportion and solve for the
unknown side.
Review
E
Similar Shapes
28
B
C
6
x
14
A
D
 ABC ~  EDC
Since we are told that  ABC ~  EDC, we also
know that AB ~ ED, BC ~ DC, and AC ~ EC
Solving
Word
Problems
1. Draw a picture/diagram
2. Make a list of what you know
and what you are looking for
3. Solve the problem
Word
Similar Shapes
Problem
At a given time of day, a building
x
20
8
4
of unknown height casts a shadow
that is 24 feet long. At the same
time of day, a post that is 8 feet tall
casts a shadow that is 4 feet long.
What is the height of the building?
You try
Workbook
p 189
# all
p 190
# 1-4