1. Pre-Algebra T McDowell
Chapter 6
Pre-Algebra
T McDowell

2. 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 4 8 =
1:3 = 3:9
- Similar describes things which have the same shape but are not the same size.
Similar

3. = If a/b = c/d then ad = bc a c b d ad = bc 2/3 = 4/6 10/x = 6/3
Cross
Multiplying
If a/b = c/d then ad = bc
a c
b d =
ad = bc
2/3 = 4/6
10/x = 6/3
5/6 = x/72
Examples

4. 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 width of the smaller figure to the larger figure:
8 ft
4 ft
2 ft
4 ft
2 ft =
8 ft
4 ft

5. 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?
height
width
4 ft
2 ft =
8 ft
x ft
8 feet
4 feet
Then solve for x by cross multiplying:
Due to the math it does not make a difference whether the smaller side is the numerator or denominator. The only thing which matters is that it is consistent on both sides of the equation.
4x = 16
X = 4
2 feet
? feet

6. 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.

7. These two stick figures are similar.
You Try
These two stick figures are similar.
Write a proportion relating the similar shapes.
Find the missing width.
8 feet
12 feet
Knowing the two figures are similar the proportion between the two stick figures is 8 feet:12 feet.
Once written as a fraction 8/12 reduces to 2/3. So the proportion between the two stick figures is 2:3.
If the proportion is 2:3 then the student should set up this equation and solve for x:
2 / 3 = 4 / x
2 * x = 3 * 4
x = 12 / 2
x = 6 feet
4 feet
x feet

8. These two trapezoids are similar.
You Try
These two trapezoids are similar.
Write a proportion relating the similar shapes.
Find the missing sides. 15 a 10 40
Discuss proportions based on this slide – tall/thin, short/broad etc. 24 x

9. You Try
The scale of a map is 1 inch : 10 miles. Find the actual distance given the distance on the map.
4 inches
10 inches
1 foot
5.5 inches
6.75 inches
Discuss proportions based on this slide – tall/thin, short/broad etc.

10. Leonardo
da Vinci
Discuss proportions based on this slide – tall/thin, short/broad etc.

11. Write a ratio that represents each statement.
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.
7 head heights
1 body height
1 wingspan
1 body height
1 foot length
1 forearm length

12. Estimated total height Wingspan
da Vinci
Proportions
Activity
Measure in inches
Head Height
Estimated total height
Wingspan
Actual height
Foot length
Estimated
forearm length
Actual forearm length

13. 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.

14. Ratios, Decimals, and Percents 11/16
A ratio that compares a number to 100
54/100 = 54%
36/100 = 36%
4/25 = 16/100 = 16%
Examples

15. Percents can also be converted into fractions.
Percents as Fractions
Percents can also be converted into fractions.
Place the percent over 100
Reduce the fraction into simplest form.
Example
88% 88
100
 4
 4 22 25

16. 7 You try Write each fraction as a percent 67/100 ¾ 32/50
Write each percent as a fraction
92%
48%
326% 7

17. Since percents can be written as fractions, they can also be converted to decimals
Percents as decimals
The fastest way to convert a percent to a decimal is to move the decimal 2 hops left.
Examples
37%
37.%
0.37

18. 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%

19. You try
Convert each percent into a decimal
25%
457%
0.4%
Convert each decimal into a percent
0.89
0.056
9.97

20. You try
Workbook
P 101
# all

21. Turn in homework
Sharpen pencil
Sit down
Get ready for notes

22. Proportions and Percents 11/17
One way to solve problems involving percents is to set up a proportion.
Proportions
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
Solve the proportion by cross multiplying

23. Write a proportion and solve. 23% of 158 15% of 24 345% of 106
You try
Write a proportion and solve.
23% of 158
15% of 24
345% of 106

24. You know you are looking for a percent or x/100
Finding the percent
What percent of 86 is 4?
You know you are looking for a percent or x/100
x = 4
Solve the proportion by cross multiplying

25. Write a proportion and solve. What percent is 56 of 109?
You try
Write a proportion and solve.
What percent is 56 of 109?
What percent is 3 of 9?
What percent is 150 of 80?

26. You have the percent so write that as a ratio: 67/100
Finding the whole amount
34 is 67% of what number
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 x
Solve the proportion by cross multiplying

27. Write a proportion and solve. 12 is 56% of what number?
You try
Write a proportion and solve.
12 is 56% of what number?
54 is 120% of what number?
21 is 5% of what number?

28. You have the percent so write that as a ratio: 15/100
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 x
Solve the proportion by cross multiplying

29. You try
Workbook
P 103
# all

30. Turn in homework
Get your workbook
Sharpen pencil
Sit down
Get ready for notes

31. 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

32. Write an equation to solve What is 14% of 65? What is 135% of 15?
You try
Write an equation to solve
What is 14% of 65?
What is 135% of 15?
What is 82% of 110?

33. Of means to multiply
Is means equal sign
Finding
the Percent
20 is what percent of 120?
20 = x • 120
 120
 120
0.167 = x
16.7% = x

34. Write a proportion and solve. 12 is what percent of 90?
You try
Write a proportion and solve.
12 is what percent of 90?
90 is what percent of 82?
34 is what percent of 150?

35. Of means to multiply
Is means equal sign
Finding
The whole
15 is 45% of what number?
15 = 45% • x
15 = 0.45 • x
 0.45
 0.45
33.3 = x

36. Write a proportion and solve. 24 is 42% of what number?
You try
Write a proportion and solve.
24 is 42% of what number?
145 is 110% of what number?
5 is 30% of what number?

37. You try
Workbook
P 105
# all

38. Binder Check What was the topic for the notes given on 11/18?
What was the answer to number 55 from the homework assigned 11/16, p 313, # 55-70
Write the calculator policy from the Classroom Guidelines and Procedures handout.

39. Writing Proportions 11/30 Similar Shapes34 10 26 x
Write a proportion and solve for the unknown side.

40. Similar Shapes Review E 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

41. Draw a picture/diagram
Make a list of what you know and what you are looking for
Solve the problem
Solving
Word
Problems

42. Similar Shapes Word Problem
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? x 8 20 4

43. You try
Workbook
p 189
# all
p 190
# 1-4