Chapter 6Pre-AlgebraT McDowell

Proportions 11/09Proportions

1

248

= 1:3 = 3:9

- If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures.

Similar - Similar describes things which have the same shape but are not the same size.

a c b d

Cross

Multiplying If a/b = c/d then ad = bc

ad =

=

1. 2/3 = 4/6

2. 10/x = 6/3

3. 5/6 = x/72

Examples

bc

A proportion can be made relating the height and the width of the smaller figure to the larger figure:

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.

Ratio

2 ft

4 ft

8 ft

4 ft4 ft

2 ft=

8 ft

4 ft

Proportion

Solving Proportion Problems

First, designate the unknown side as x. Then, set up an equation using proportions. What does the numerator represent? What does the denominator represent?

2 feet

4 feet

8 feet

? feet

4 ft

2 ft=

8 ft

x ft

4x = 16

X = 4

Then solve for x by cross multiplying:

height

width

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

1. Write a proportion relating the similar shapes.

2. Find the missing width.

4 feet

8 feet 12 feet

x feet

These two stick figures are similar.

You Try These two trapezoids are similar.

1. Write a proportion relating the similar shapes.

2. Find the missing sides.

15

x

a

24

40

10

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 Vinci1452 - 1519

The average adult human figure is about 7 to 7.5 heads tall.

The arms' wingspan (measured from the tips of the middle fingers) is about equal to the

body height.

The length of the foot is about equal to the length of the forearm.

Write a ratio that represents each statement.

7 head heights1 body height

1 wingspan1 body height

1 foot length1 forearm length

Head Height

Estimated total height

Wingspan

Estimated total height

Actual height

Foot length

Estimatedforearm length

Actual forearm length

da VinciProportionsActivity

Measure in inches

•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•bottom of nose to the bottom of the chin.

•Width of head is between four and five eyes wide. •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

Use these proportions to draw a head.

Ratios, Decimals, and Percents 11/16Percent A ratio that compares a number to

100

Examples 54/100 = 54%

36/100 = 36%

4/25 = 16/100 = 16%

Percents can also be converted into fractions.

Percents as Fractions

Place the percent over 100

Reduce the fraction into simplest form.Example 88%

88100

4 4

2225

You try Write each fraction as a percent

1. 67/100

2. ¾

3. 32/50

Write each percent as a fraction

1. 92%

2. 48%

3. 326%

7

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 as Percents

Decimals can also be converted to percents

The fastest way to convert a decimal to a percent is to move the decimal 2 hops right.

Examples 3.45

345%

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

You try

•Turn in homework

•Sharpen pencil

•Sit down

•Get ready for notes

Proportions and Percents 11/17Proportions One way to solve problems

involving percents is to set up a proportion.What is 45% of 60?

45 = x100 60

Solve the proportion by cross multiplying

We know 45% is 45/100, but we don’t know what part of 60 we need so that is x/60

Write a proportion and solve.

1. 23% of 158

2. 15% of 24

3. 345% of 106

You try

Finding the percent

What percent of 86 is 4?

You know you are looking for a percent or x/100

x = 4100 86

Solve the proportion by cross multiplying

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?

You try

Finding the whole amount

34 is 67% of what number

You have the percent so write that as a ratio: 67/100

67 = 34100 x

Solve the proportion by cross multiplying

34 is the numerator of the other ratio—we don’t know the denominator: 34/x

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?

You try

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

15 = 90100 x

Solve the proportion by cross multiplying

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

Workbook

P 103

# all

You try

•Turn in homework

•Get your workbook

•Sharpen pencil

•Sit down

•Get ready for notes

Percents and Equations 11/18Math

WordsOf means to multiply

Is means equal sign

What is 35% of 90?

x = 35% x 90x = 0.35 x 90

Finding

the part

x = 31.5

Write an equation to solve

1. What is 14% of 65?

2. What is 135% of 15?

3. What is 82% of 110?

You try

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

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?

You try

Finding The whole

Of means to multiply

Is means equal sign

15 is 45% of what number?

15 = 45% • x

0.45 0.4533.3 = x

15 = 0.45 • x

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

You try

Binder Check1. 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.

Write a proportion and solve for the unknown side.

1034

26x

Writing Proportions 11/30

Similar

Shapes

Similar ShapesReview

14

6

28

x

B

A

C

D

E

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

Similar ShapesWord Problem

x

420

8

At a given time of day, a building 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?

Workbook

p 189

# all

p 190

# 1-4

You try