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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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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:
With a colon : 6 ft : 8 ft = 6 : 8 = 3 : 4
With the word to : 6 ft to 8 ft. = 6 to 8 = 3 to 4.
Notice that in each case,
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
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
- ← Cancel common factors
(^2) x 3 7 5
(^2) x 3 7
- = • ← Multiply both sides by the reciprocal of 2/
x 21 10