Proportions, similar figures, and percents

Proportions, Similar Figures,

and PercentsAlgebra I

By Ita Rodríguez

Proportions

Proportions

• A proportion is an equation that states that two ratios (written as fractions) are equal.

=

Proportions

• The cross product of a proportion are equal.

• We can use this property to solve proportions for a missing quantity.

=

1= 2=

Problem 1:

• What is the solution to the proportion = ?

=5= 4

= ==

Use cross multiplication to solve.

Problem 2:


=

4(𝑏−8) =

5(𝑏+3)

=

Use cross multiplication to solve.

Use the distributive property on both sides of the equation.

Problem 2:


=

==

Use what you know about solving equations to finish the problem.

= 𝑏 𝑏+32 +32

=

Similar Figures

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

Similar Figures• Similar is represented by the

symbol . The sides are similar.

• c

• Congruent is represented by the symbol . The angles are congruent.

• E• •

∠𝐴∠𝐵

∠𝐶 ∠𝐸∠𝐹

∠𝐺𝑎 𝑏

𝑐 𝑔

𝑓𝑒

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 CFE

AD

16

18

12

10

𝑌

Problem 3:


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 CFE

AD

16

18

12

10

𝑌

Problem 3:


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 CFE

AD

16

18

12

10

𝑌

Problem 3:


Now you will set up the proportion using values from the smaller triangle (∆ABC) on one side andvalues from the larger triangle (∆DEF) on the other.B C

FE

AD

16

18

12

10

𝑌

= Notice that corresponds to 10 and 12 corresponds to 16.

Problem 3:


Now you will solve by cross multiplication.

=

B CFE

AD

16

18

12

10

𝑌

=

Percents

Percents

• A percent standardizes a comparison to a common base of 100.

• Percents can also be written as decimals by dividing.

25%=

=0.25

Percents

• Percent problems can be solved using both fractions or decimals. Key words can help solve them.

• Method 1: Fractions (proportions) what is up percent of down

Problem 4:

125% of what number is 17.5?

We begin by underlining and identifying key words.


KEY WORDSwhat = is = uppercent = of = down

125❑ =❑

❑

Problem 4:



KEY WORDSwhat = is = uppercent = of = down% means over

100

125100=

❑❑

Problem 4:



KEY WORDSwhat = is = uppercent = of = downThe # or word after

goes down

125100=

❑❑

Problem 4:



KEY WORDSwhat = is = uppercent = of = down𝒙

125100=

❑𝑥

Problem 4:



KEY WORDSwhat = is = uppercent = of = downThe # or word after

goes up

125100=

17.5𝑥

Problem 4:

125% of what number is 17.5?Now you can solve by cross multiplication.

=125 17.5125

=

𝑥=14

Percents

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

Problem 5:

What is 20% of 140?

We begin by underlining and identifying key words. What is 20% of 140?

KEY WORDSwhat : is : =percent : decimal of : multiply

Problem 5:

What is 20% of 140?

What is 20% of 140? =


Problem 5:

What is 20% of 140?

What is 20% of 140? = 0.20


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

Problem 5:

What is 20% of 140?

What is 20% of 140? = 0.20


Problem 5:

What is 20% of 140?

What is 20% of 140? = 0.20140


Problem 5:

What is 20% of 140?Now we can solve the equation. = 0.20140 = 28

The End

Proportions, similar figures, and percents

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