Topic D: Percents , Ratios, Rates, and Proportions
D1. Percents
30
= 0.3 . By this
100
definition, we can convert between percentages through the following procedures.
The word percentage literally means "per 100", so 30% means 30 per 100, or
•
•
To convert from percentage to decimal: divide by 100 (and remove the "%" sign).
To convert from decimal to percentage: multiply by 100 (and add a "%" sign).
From our discussion on place value, when we
multiply by 100 it has the effect of moving the
decimal point 2 places the right. When we divide
by 100 we move the decimal point 2 places to the
left.
D2. Percents and Ratios
At basketball camp, Carlos made 13 free throws out of 25 attempts. In mathematics, we say
“The ratio of the number of free throws made to the number Carlos attempted is 13 to 25” and
write 13:25 or 13/25. The scenario presented by Carlos can be represented by three different
ratios.
• 13 free throws to 25 attempts or 13:25 (part-whole ratio)
• 13 made free throws to 12 missed free throws or 13: 12 (part-part ratio)
• 12 missed free throws to 25 attempts or 12:25 (part-whole ratio)
A ratio is closely linked to our ideas of rational numbers and we can express these part-whole
ratios as percentages. That is, Carlos made (13/25 = 0.52) 52% of his free throws at basketball
camp.
For more practice, please visit the Percent as Ratios Learning Object
D3 Percents and Proportions
A proportion is a statement that two ratios are equal. It can be expressed as two equal fractions
4 20
=
. This proportional statement is read as "the ratio twenty to twenty-five is equal to the
5 25
ratio four is to five."
Definition: If
a
c
a c
and are two ratios, then the statement of equality = is called a
b
d
b d
proportion.
To ensure that two ratios are equal, we use a process called cross multiplication.
Example
4 20
=
since
5 25
5 × 20 = 25 × 4
Theorem: Condition for Proportions
a c
The equality = is a proportion if and only if ad = bc.
b d
4 20
=
5 25
a c
=
b d
To solve percent problems, we can also use this cross multiplication to find a missing term
in the appropriate proportion. This method will help us solve the following percent as proportion
problems.
• What is 35% of 240?
• 64% of what number is 120?
• What percent of 420 is 240?
What is 35% of 240?
SHORTCUT
What is 35% of 240?
x = 0.35 x 240
x = 84
35
x
=
100 240
100x = 35 × 240
Æ 100x = 8400
x=
8400
= 84
100
64% of what number is 120?
SHORTCUT
64% of what number is 120?
0.64x = 120
64 100
=
120
x
64x = 120 × 100
Æ 64x = 12,000
x=
12,000
= 187.5
64
What percent of 420 is 240?
120
x=
= 187.5
0.64
SHORTCUT
What percent of 420 is 240?
420x = 240
x
100
=
240 420
420x = 240 × 100
Æ 420x = 24,000
x=
24,000
≈ 57.2
420
So 57.2% of 420 is 240.
x=
240
≈ 0.572
420
So 57.2% of 420 is 240.
To examine how to solve these types of percent problems, please visit the Percent and
Proportions Learning Object.
D4. Proportional Reasoning
To solve proportional reasoning problems, we can also use this cross multiplication to find
a missing term in a proportion. Here's an example. In a horror movie featuring a giant beetle, the
beetle appeared to be 50 feet long. However, a model was used for the beetle that was really only
20 inches long. A 30-inch tall model building was also used in the movie. How tall did the
building seem in the movie?
Solution:
length of beetle in movie height of building in movie
=
length of beetle in model height of building in model
50
x
=
20 30
20x = 50 x 30
20x = 1500
x = 75 feet
To practice solving problems with proportions, please visit the following Learning Objects: