RATIO RATES AND PROPORTIONS

Ratio and Proportion

A ratio is a comparison of two quantities by division. The ratio of a to b can be written a : b or a/b, where b ≠ 0.

Ratios that name the same comparison are said to be equivalent. A statement that two ratios are equivalent, such as 1/12 = 2/24 , is called a proportion.

Using Ratios

Example 1 :

The ratio of faculty members to students at a college is 1:15. There are 750 students. How many faculty members are there?

Solution :

Write a ratio comparing faculty to students.

Faculty ----> 1

Students ----> 15

Write a proportion. Let x be the number of faculty members.

x/750 = 1/15

Because x is divided by 750, multiply each side of the equation by 750.

750 ⋅ (x/750) = (1/15) ⋅ 750

x = 50

There are 45 faculty members.

Rates

A rate is a ratio of two quantities with different units, such as 48 mi/3 gal. Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as 16 mi/1 gal, or 16 mi/gal. You can convert any rate to a unit rate.

Finding Unit Rates

Example 2 :

David ate 58.5 hot dogs in 13 minutes to win a contest. Find the unit rate in hot dogs per minute. Round to the nearest hundredth.

Solution :

Write a proportion to find an equivalent ratio with a second quantity of 1.

58.5/13 = x/1

4.5 = x

The unit rate is approximately 4.46 hot dogs per minute.

Dimensional Analysis

Dimensional analysis is a process that uses rates to convert measurements from one unit to another. A rate such as 12 in./1 ft, in which the two quantities are equal but use different units, is called a conversion factor . To convert from one set of units to another, multiply by a conversion factor.

Using Dimensional Analysis

Example 3 :

A large adult male human has about 12 pints of blood. Use dimensional analysis to convert this quantity to gallons.

Solution :

Step 1 :

Convert pints to quarts.

Multiply by a conversion factor whose first quantity is quarts and whose second quantity is pints.

12 pt ⋅ (1 qt/2 pt) = 6 qt

12 pints is 6 quarts.

Step 2 :

Convert quarts to gallons.

Multiply by a conversion factor whose first quantity is gallons and whose second quantity is quarts.

6 qt ⋅ (1 gal/4 qt) = 6/4 gal = 1½ gal

A large adult male human has about 1½ gallons of blood.

Cross Products

In the proportion a/b = c/d, the products a · d and b · c are called cross products.

You can solve a proportion for a missing value by using the Cross Products Property.

Cross Products Property

Words :

In a proportion, cross products are equal.

Numbers :

Algebra :

Solving Proportions

Solve each proportion.

Example 4 :

7/9 = 3/x

Solution :

7/9 = 3/x

Use cross products.

7x = 3(9)

7x = 27

Divide each side by 7.

x = 27/7

Example 5 :

5/9 = 10/(x + 2)

Solution :

5/9 = 10/(x + 2)

Use cross products.

5(x + 2) = 9(10)

5x + 10 = 90

Subtract 10 from each side.

5x = 80

Divide each side by 5.

x = 16

Scale Drawings and Scale Models

A scale is a ratio between two sets of measurements, such as 1 in : 7 mi. A scale drawing or scale model uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing.

Example 6 :

On a map with a scale of 1 in = 18 mi, the distance from Chicago to Evanston is 0.63 in. What is the actual distance ?

Solution :

Write the scale as a fraction.

map/actual = 1 in/18 mi

Let x be the actual distance.

1/18 = 0.63/x

Use cross products to solve.

1(x) = 18(0.63)

x = 11.34 miles

The actual distance is 11.34 miles.

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