This presentation will help algebra students learn how to solve proportions using cross multiplication. They will also be able to use cross multiplication to find a missing side with similar figures. Last, they will be able to solve percents using proportions and decimal equations.

1. Proportions, Similar
Figures, and Percents
Algebra I
By Ita Rodríguez

2. Proportions

3. Proportions
• A proportion is an equation that states that two ratios (written as fractions)
are equal.
1
2
=
8
16

4. Proportions
• The cross product of a proportion are equal.
• We can use this property to solve proportions for a missing quantity.
1
2
=
8
16
1∙ 16 = 2∙ 8
16 = 16

5. Problem 1:
• What is the solution to the proportion
𝑦
7
=
4
5
?
𝑦
7
=
4
5
5∙ 𝑦 = 4∙ 7
5𝑦 = 28
5𝑦
5
=
28
5
𝑦 = 5.6
Use cross multiplication to solve.

6. Problem 2:
• What is the solution to the proportion
𝑏−8
5
=
𝑏+3
4
?
𝑏−8
5
=
𝑏+3
4
4(𝑏−8) =5(𝑏 + 3)
4𝑏−32 = 5𝑏 + 15
Use cross multiplication to solve.
Use the distributive property on
both sides of the equation.

7. Problem 2:
• What is the solution to the proportion
𝑏−8
5
=
𝑏+3
4
?
−1𝑏−32 = 15
−1𝑏
−1
=
47
−1
𝑦 = −47
Use what you know about solving
equations to finish the problem.
4𝑏−32 = 5𝑏 + 15
−5𝑏 −5𝑏
+32 +32
−1𝑏 = 47

8. Similar Figures

9. Similar Figures
• Figures are similar if the have the same shape, but not necessarily the same
size.
• The angles are congruent (measure the same), but the side lengths are not.
∠𝐴
∠𝐵
∠𝐶 ∠𝐸
∠𝐹
∠𝐺

10. Similar Figures
• Similar is represented by the symbol ~.
The sides are similar.
• 𝑎~𝑒
• 𝑏~𝑓
• c~𝑔
• Congruent is represented by the symbol
≅. The angles are congruent.
• ∠𝐴 ≅ ∠E
• ∠𝐵 ≅ ∠F
• ∠𝐶 ≅ ∠G
∠𝐴
∠𝐵
∠𝐶 ∠𝐸
∠𝐹
∠𝐺
𝑎 𝑏
𝑐 𝑔
𝑓𝑒

11. Problem 3:
In the diagram ∆ABC~∆DEF. What is AB?
What is the question asking?
AB represents a side, so the question is asking you
find the length of the side AB.
Write in a variable for the side you are trying to find.
B C
FE
A
D
16
18
12
10𝑌

12. Problem 3:
In the diagram ∆ABC~∆DEF. What is AB?
You will need to set up a proportion using corresponding sides.
If you look at the original question, it tells you that ∆ABC~∆DEF. The order the letters are written in
have significance.
This means that A corresponds to D (∆ABC~∆DEF),
B corresponds to E (∆ABC~∆DEF),
and C corresponds to F (∆ABC~∆DEF).
B C
FE
A
D
16
18
12
10𝑌

13. Problem 3:
In the diagram ∆ABC~∆DEF. What is AB?
Side AB corresponds to side DE.
Side BC corresponds to side EF.
Side AC corresponds to side DF.
Since DF does not have a length written, it is not useful. Therefore, you will not need to use the
correspondence between side AC and side DF.
B C
FE
A
D
16
18
12
10𝑌

14. Problem 3:
In the diagram ∆ABC~∆DEF. What is AB?
Now you will set up the proportion using values
from the smaller triangle (∆ABC) on one side and
values from the larger triangle (∆DEF) on the other.
B C
FE
A
D
16
18
12
10𝑌
𝑦
12
=
10
16
Notice that 𝑦 corresponds to 10 and 12
corresponds to 16.

15. Problem 3:
In the diagram ∆ABC~∆DEF. What is AB?
Now you will solve by cross multiplication.
16𝑦 = 12 ∙ 10
16𝑦 = 120
16𝑦
16
=
120
16
𝑦 = 7.5
B C
FE
A
D
16
18
12
10𝑌
𝑦
12
=
10
16

16. Percents

17. Percents
• A percent standardizes a comparison to a common base of 100.
• Percents can also be written as decimals by dividing.
25%=
25
100
25
100
=0.25

18. Percents
• Percent problems can be solved using both fractions or decimals. Key words
can help solve them.
• Method 1: Fractions (proportions)
what 𝑥 is up
percent
100
of down

19. Problem 4:
125% of what number is 17.5?
We begin by underlining and identifying key words.
125% of what number is 17.5?
KEY WORDS
what = 𝑥 is = up
percent =
100
of = down
125
=

20. Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDS
what = 𝑥 is = up
percent =
100
of = down
% means over 100
125
100
=

21. Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDS
what = 𝑥 is = up
percent =
100
of = down
The # or word after goes down
125
100
=

22. Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDS
what = 𝑥 is = up
percent =
100
of = down
𝒙
125
100
=
𝑥

23. Problem 4:
125% of what number is 17.5?
125% of what number is 17.5?
KEY WORDS
what = 𝑥 is = up
percent =
100
of = down
The # or word after goes up
125
100
=
17.5
𝑥

24. Problem 4:
125% of what number is 17.5?
Now you can solve by cross multiplication.
125
100
=
17.5
𝑥
125𝑥 =17.5∙ 100
125𝑥 = 1,750
125𝑥
125
=
1,750
125
𝑥 = 14

25. Percents
• Method 2: Decimals
• what 𝑥 is =
percent convert the number into of multiply
a decimal by moving the
decimal point two places
to the left

26. Problem 5:
What is 20% of 140?
We begin by underlining and identifying key words.
What is 20% of 140?
𝑥
KEY WORDS
what : 𝑥 is : =
percent : decimal of : multiply

27. Problem 5:
What is 20% of 140?
What is 20% of 140?
𝑥 =
KEY WORDS
what : 𝑥 is : =
percent : decimal of : multiply

28. Problem 5:
What is 20% of 140?
What is 20% of 140?
𝑥 = 0.20
KEY WORDS
what : 𝑥 is : =
percent : decimal of : multiply
20 is written as a decimal by moving the decimal point that
is implied behind the zero (20.) two places to the left: 0.20

29. Problem 5:
What is 20% of 140?
What is 20% of 140?
𝑥 = 0.20 ∙
KEY WORDS
what : 𝑥 is : =
percent : decimal of : multiply

30. Problem 5:
What is 20% of 140?
What is 20% of 140?
𝑥 = 0.20∙140
KEY WORDS
what : 𝑥 is : =
percent : decimal of : multiply

31. Problem 5:
What is 20% of 140?
Now we can solve the equation.
𝑥 = 0.20∙140
𝑥 = 28

32. The End