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. 

Answers

1. Answer :

322 miles/11.5 gallons  =  28 miles/gallon

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

2. Answer :

Density  =  Mass/Volume

=  483 grams/46 cm³

=  10.5 grams/cm³

The density of silver is 10.5 grams per cubic centimeter.  

3. Answer :

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

x, 2x, 3x 

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

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°

4. Answer :

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

5. Answer :

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

6. Answer :

From the ratio 6 : 7, 

length  =  6x

width  =  7x

Given : Area of the rectangle is 378 square meters. 

l ⋅ 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.

7. Answer :

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. 

8. Answer :

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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