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Ratios, Rates, and Proportions: A Comprehensive Guide, Slides of Pre-Calculus

Proportions are often used to solve a variety of problems, such as estimating wildlife populations, scaling distances on a map, or calculating mixtures and ...

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RATIO AND PROPORTION
RATIOS AND RATES
Quantities such as 8 feet, 16 cents or 10 hours are numerical quantities written with units. A ratio is a
comparison of two quantities with the same units. For example, if we want to compare the heights of
two trees, one 6 feet tall and the other 8 feet tall, we can write this ratio three ways:
1) As a fraction: 6ft
8ft
62
8
== 3
2
3
4
4
=
2) With a colon: 6 ft : 8 ft = 6 : 8 = 3 : 4
3) With the word to: 6 ft to 8 ft. = 6 to 8 = 3 to 4.
Notice that in each case,
The order of the numbers in a ratio is important! To write a ratio as a fraction, place the first
number of the ratio in the numerator; place the second number in the denominator.
The ratio is written in lowest terms.
The units are not written as part of the ratio. Because a ratio compares two quantities with the
same units – convert the units if they are different.
Example 1: Write each ratio as a fraction in lowest terms.
A) 15 pounds to 24 pounds
The units are the same, so we can write this ratio as
15 pounds to 24 pounds = 15 3
24 =5
3
5
8
8
=
B) 75 cents to $1.25
Here, the units are not the same. Since $1 = 100 cents, to convert $1.25 to cents, drop the
dollar sign and move the decimal point two places to the right.
$1.25 = 1.25 = 125 cents
75 cents to $1.25 = 75 cents to 125 cents = 75 3 25
125 =
525
3
5
=
A rate is a comparison of two quantities with different units, such as 10 g per 180 mL. Like a ratio, a
rate can be written as a fraction, with a colon, or with the word to. A rate is also expressed in lowest
terms. Unlike a ratio, the units are written as part of the rate. For example, to write the rate “10 g per 180 mL” as a
fraction in lowest terms, cancel the corresponding 0’s and keep the units:
10 g per 180 mL = 10 g
18 0
1 g
18 mL
mL =
PBCC 1 SLC Lake Worth Math Lab
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RATIO AND PROPORTION

R ATIOS AND R ATES

Quantities such as 8 feet, 16 cents or 10 hours are numerical quantities written with units. A ratio is a comparison of two quantities with the same units. For example, if we want to compare the heights of two trees, one 6 feet tall and the other 8 feet tall, we can write this ratio three ways:

  1. As a fraction : 6 ft 8 ft
  1. With a colon : 6 ft : 8 ft = 6 : 8 = 3 : 4

  2. With the word to : 6 ft to 8 ft. = 6 to 8 = 3 to 4.

Notice that in each case,

  • The order of the numbers in a ratio is important! To write a ratio as a fraction, place the first number of the ratio in the numerator ; place the second number in the denominator.
  • The ratio is written in lowest terms.
  • The units are not written as part of the ratio. Because a ratio compares two quantities with the same units – convert the units if they are different.

Example 1: Write each ratio as a fraction in lowest terms.

A) 15 pounds to 24 pounds

The units are the same, so we can write this ratio as

15 pounds to 24 pounds =

- 5

B) 75 cents to $1.

Here, the units are not the same. Since $1 = 100 cents, to convert $1.25 to cents, drop the dollar sign and move the decimal point two places to the right.

$1.25 = 1.25 = 125 cents

75 cents to $1.25 = 75 cents to 125 cents = 75 3 25 125

A rate is a comparison of two quantities with different units, such as 10 g per 180 mL. Like a ratio, a rate can be written as a fraction, with a colon, or with the word to. A rate is also expressed in lowest terms. Unlike a ratio, the units are written as part of the rate. For example, to write the rate “10 g per 180 mL” as a fraction in lowest terms, cancel the corresponding 0’s and keep the units:

10 g per 180 mL = 10 g 18 0

1 g mL 18 mL

PROPORTIONS

A proportion is a mathematical statement that two ratios or rates are equal. For example, whenever we write equivalent fractions, we create a proportion, such as the one shown below:

3 6 4 8

In a true proportion, the cross products are equal :

3 6 4 8

Because the cross products are equal, we can solve a proportion when one of the numbers is unknown.

Example 2: Solve 4 12 9 x

To solve the proportion 1) cross multiply the ratios, 2) write an equation; and 3) solve for the variable.

← Cross multiply the ratios

4 • x = 9 • 12 ← Write an equation

9 x

4x = 108 4 x 4

← Solve the equation for x x = 27

Example 3: Solve

4 5 x 2 3 7 4

=

To solve the proportion, begin by finding the cross products:

5 x 2 3 7 4

=

4

(^2) x 4 7

• =^3

- ← Cancel common factors

(^2) x 3 7 5

(^2) x 3 7

- = ← Multiply both sides by the reciprocal of 2/

x 21 10