# USING RATIOS AND RATES TO SOLVE PROBLEMS WORKSHEET

Problem 1 :

A car travels 322 miles on 11.5 gallons of gas. What is the car’s gas mileage ?

Problem 2 :

A volume of 46 cm3 of silver has a mass of 483 grams. What is the density of silver ?

Problem 3 :

If the angles of a triangle are in the ratio of 1 : 2 : 3, then find the measure of each angle.

Problem 4 :

The sum of two numbers is 14 and the ratio of the two numbers is -3. What is the product of the two numbers ?

Problem 5 :

Let 5(x + y)  =  7y. Find the ratio between x and y.

Problem 6 :

The length and width of a rectangular garden are in the ratio 6 : 7. If the area of the garden is 378 square meters, find  perimeter of the garden.

Problem 7 :

Mr. John can travel 218.5 miles of distance on consumption of 9.5 gallons of gas. How many miles can he travel on 15.5 gallons of gas ?

Problem 8 :

The ratio between the speeds of two cars is 7 : 8. If the second car runs 400 miles in 5 hours, then find the speed of the first car.

## Detailed Answer Key

Problem 1 :

A car travels 322 miles on 11.5 gallons of gas. What is the car’s gas mileage ?

Solution :

322 miles / 11.5 gallons  =  28 miles / gallon

The car's gas mileage is 28 miles per gallon.

Problem 2 :

A volume of 46 cm3 of silver has a mass of 483 grams. What is the density of silver ?

Solution :

Density  =  Mass / Volume

Density  =  483 grams / 46 cm3

Density  =  10.5 grams / cm3

The density of silver is 10.5 grams per cubic centimeter.

Problem 3 :

If the angles of a triangle are in the ratio of 1 : 2 : 3, then find the measure of each angle.

Solution :

From the ratio 1 : 2 : 3, the three angles are

x, 2x, 3x

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

Then,

x + 2x + 3x  =  180°

6x  =  180

Divide each side by 6.

x  =  30

Measure of each angle :

First angle  =  30°

Second angle  =  2(30)  =  60°

Third angle  =  3(30)  =  90°

Problem 4 :

The sum of two numbers is 14 and the ratio of the two numbers is -3. What is the product of the two numbers ?

Solution :

Let x and y be the numbers.

The sum of two numbers is 14.

x + y  =  14 -----(1)

Ratio of the two numbers is -3.

x/y  =  -3

Multiply each side by y.

x  =  -3y -----(2)

Substitute -3y for x in (1).

-3y + y  =  14

-2y  =  14

Divide each side by -2.

y  =  -7

Substitute -7 for y in (2).

x  =  -3(-7)

x  =  21

So,, the two numbers are 21 and -7.

Product of the two numbers :

=  21(-7)

=  -147

Problem 5 :

Let 5(x + y)  =  7y. Find the ratio between x and y.

Solution :

5(x + y)  =  7y

5x + 5y  =  7y

Subtract 5y to each side.

5x  =  2y

Divide each side by 5y.

5x/5y  =  2y/5y

x/y  =  2/5

x : y  =  2 : 5

Problem 6 :

The length and width of a rectangular garden are in the ratio 6 : 7. If the area of the garden is 378 square meters, find  perimeter of the garden.

Solution :

From the ratio 6 : 7,

length  =  6x

width  =  7x

Given : Area of the rectangle is 378 square meters.

w  =  378

Substitute.

6x ⋅ 7x  =  378

42x2  =  378

42x2  =  378

Divide each side by 42.

x2  =  9

Take square root on both sides.

x  =  3

Find the length and width :

length  =  6(3)  =  18 meters

width  =  7(3)  =  21 meters

Perimeter of the rectangle :

=  2l + 2w

Substitute.

=  2(18) + 2(21)

=  36 + 42

=  78

The perimeter of the garden is 78 meters.

Problem 7 :

Mr. John can travel 218.5 miles of distance on consumption of 9.5 gallons of gas. How many miles can he travel on 15.5 gallons of gas ?

Solution :

Distance covered on 9.5 gallons of gas  =  218.5 miles

Distance covered on 1 gallon of gas :

=  218.5 / 9.5

=  23 miles

Distance covered on 15.5 gallons of gas :

=  15.5 ⋅ 23

=  356.5 miles

Mr. John can travel 356.5 miles on 15.5 gallons of gas.

Problem 8 :

The ratio between the speeds of two cars is 7 : 8. If the second car runs 400 miles in 5 hours, then find the speed of the first car.

Solution :

From the ratio 7 : 8,

speed of the first car  =  7x

speed of the second car  =  8x

Given : The second car runs 400 miles in 5 hours.

Time  =  Distance / Speed

Substitute.

5  =  400 / 8x

5  =  50 / x

Multiply each side by x.

5x  =  50

Divide each side by 5.

x  =  10

Speed of the first car :

=  7(10)

=  70 miles per hour

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