APPLYING RATIOS AND RATES

Comparison of two quantities of same kind with same units is represented as a ratio.

For example, comparison of number of boys and girls in a school is represented as ratio between number of boys and girls.

On the other hand, comparison of the given measure to one unit of another measure is represented as rate. 

For example, number of miles covered by a car in one hour is represented as rate.

A unit rate is a rate that has a denominator of 1. Some examples of unit rates are defined as follows.

Example 1 :

1360 grams of coffee cost $17.68. What is the unit price of the coffee per gram?

Solution :

$17.68/1360 grams  =  1768 cents/1360 grams

$17.68/1360 grams  =  1.3 cents/gram

The unit price of the coffee is 1.3 cents per gram.

Example 2 :

At a grocery store, 20 fl oz of brand A vitamin water is sold for $0.95. What is the unit price of the vitamin water per ounce, to the nearest cents?

Solution :

$0.95/20 fl oz  =  95 cents/20 fl oz

$0.95/20 fl oz  =  4.75 cents/fl oz

The unit price of the the vitamin water per ounce is about 5 cents.

Example 3 :

Three angles of a triangle are in the ratio of 3 : 5 : 7. What is the measure of each angle?

Solution :

From the ratio 3 : 5 : 7, the three angles are

2x, 7x, 11x 

In any triangle, sum of the angles is equal to 180°.

Then,

2x + 7x + 11x  =  180°

20x  =  180

Divide each side by 20.

x  =  9

Measure of each angle :

First angle  =  2(9)  =  18°

Second angle  =  7(9)  =  63°

Third angle  =  11(9)  =  99°

Example 4 :

The ratio of 7/4 to 5/2 is equal to the ratio 14 to what number?

Solution :

Let x be the required number.

Then,

7/4 : 5/2  =  14 : x

Write the above ratios as divisions.

7/4 ÷ 5/2  =  14 ÷ x

(7/4) ⋅ (2/5)  =  14/x

7/10  =  14/x

7x  =  14 ⋅ 10

7x  =  140

Divide each side by 7.

x  =  20

So, the required number is 20.

Example 5 :

If 2(x - y)  =  3y, what is the ratio x/y?

Solution :

2(x - y)  =  3y

2x - 2y  =  3y

Add 2y to each side.

2x  =  5y

Divide each side by 2y.

2x/2y  =  5y/2y

x/y  =  5/2

Example 6 :

The length and width of a rectangular garden are in the ratio 6 : 7. If the perimeter of the rectangle is 78 meters, what is the area of the garden in square meters?

Solution :

From the ratio 6 : 7,

length  =  6x

width  =  7x

Given : Perimeter of the rectangle is 78 meters.

2l + 2w  =  78

Substitute.

2(6x) + 2(7x)  =  78

12x + 14x  =  78

26x  =  78

Divide each side by 26.

x  =  3

Find the length and width :

length  =  6(3)  =  18 meters

width  =  7(3)  =  21 meters

Area of the rectangle :

=  l ⋅ w

Substitute.

=  18 ⋅ 21

=  378

The area of the garden 378 square meters.

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