RATIO RATES AND PROPORTIONS WORD PROBLEMS

Problem 1 :

The ratio of the number of boys to the number of girls in a school of 720 students is 3 : 5. Find the number of boys and girls.

Solution :

Let x be the actual length of the car.

Given : The ratio of the number of boys to the number of girls is 3 : 5.

Number of boys = 3x

Number of girls = 5x

Given : There are 720 students in school.

3x + 5x = 720

8x = 720

x = 90

Number of boys = 3(90) = 270

Number of girls = 5(90) = 450

Problem 2 :

A model car is scaled so that 1 inch of the model equals 6 feet of the actual car. If the model is 1⅔ inch long. How long is the actual car?

Solution :

Let x be the actual length of the car.

From the given information,

1 : 6 = 1⅔ : x

1 : 6 = ⁵⁄₃ : x

⅙ = ⁵⁄₃ₓ

3x = 30

x = 10

The actual length of the car is 10 feet.

Problem 3 :

A car is traveling at a constant rate of 54 miles per hour. How many kilometers will the car travel in 5 minutes? (1 mile = 1.6 kilometers)

Solution :

1 hour ----> 54 miles

1 hour ----> 54 ⋅ 1.6 kilometers

1 hour ----> 86.4 kilometers

60 minutes ----> 86.4 kilometers

1 minute ----> ⁸⁶^.⁴⁄₆₀ kilometers

1 minute ----> 1.44 kilometers

5 minutes ----> 5 ⋅ 1.44 kilometers

5 minutes ----> 7.2 kilometers

The car will travel 7.2 kilometers in 5 minutes.

Problem 4 :

The ratio of males to females in an office is 6 : 7. If there are 42 males in the office, what is the total number of people in the office?

Solution :

Given : The ratio of males to females in an office is 6 : 7.

Number of males = 6x

Number of females = 7x

Given : There are 42 males.

6x = 42

x = 7

Total number of people in the office :

= 6x + 7x

= 13x

Substitute x = 7.

= 13(7)

= 91

Problem 5 :

If 20 machines produce 1,240 printers in a day, how many more machines are needed to produce 1,984 printers in a day?

Solution :

Let x be the number of printers required to produce 1,984 printers in a day.

From the given information,

20 : 1240 = x : 1984

²⁰⁄₁₂₄₀ = ˣ⁄₁₉₈₄ 

1984(²⁰⁄₁₂₄₀) = x

32 = x

Altogether we need 32 machines. Already we have 20 machine.

Number of additional machines required :

= 32 - 20

= 12

12 more machines are needed to produce 1,984 printers in a day.

Problem 6 :

If ¾ quart of lemonade concentrate is mixed with 6⅔ quarts of water to make lemonade for 40 people, how many quarts of lemonade concentrate are needed to make the lemonade for 24 people?

Solution :

In this problem, "6⅔ quarts of water" is unnecessary information.

Let x be the number of quarts of lemonade concentrate needed to make the lemonade for 24 people.

From the given information,

¾ : 40 = x : 24

³⁄₁₆₀ = ˣ⁄₂₄

24(³⁄₁₆₀) = x

⁹⁄₂₀ = x

⁹⁄₂₀ quart of lemonade concentrate needed to make the lemonade for 24 people.

Problem 7 :

A machine produced 735 tapes in 5¼ hours What fraction of the 735 tapes was produced in one hour?

Solution :

5¼ hours ----> 735 tapes

²¹⁄₄ hours ----> 735 tapes

1 hour ----> (⁴⁄₂₁)735 tapes

1 hour ----> ⁴⁄₂₁ of 735 tapes

⁴⁄₂₁ of 735 tapes was produced in one hour.

Problem 8 :

A 32-acre field yields 768 bushels of corn each year. How many more acres are needed to yield 960 bushels of corn each year?

Solution :

Let x be the number of acres needed to yield 960 bushels of corn each year.

From the given information,

32 : 768 = x : 960

³²⁄₇₆₈ = ˣ⁄₉₆₀

960(³²⁄₇₆₈) = x

40 = x

40 acres are needed to yield 960 bushels of corn each year.

Problem 9 :

The length of a rectangle is 8 inches longer than the width. If the ratio of the length to perimeter is 5:16, what is the area of the rectangle?

Solution :

Let x be the width of the rectangle. Then the lenth of the rectangle is (x + 8) inches.

Perimeter of the rectangle :

= 2(length + width)

= 2[(x + 8) + x]

= 2(2x + 8)

= 4x + 16

Length : Perimeter = 5 : 16

(x + 8) : (4x + 16) = 5 : 16

⁽ˣ ⁺ ⁸⁾⁄₍₄ₓ ₊ ₁₆₎ = ⁵⁄₁₆

16(x + 8) = 5(4x + 16)

16x + 128 = 20x + 80

48 = 4x

12 = x

width = 12 inches

length = 20 inches

Area of the rectangle :

= length ⋅ width

= 12 ⋅ 20

= 240 in²

Problem 10 :

If 12 grams of coffee costs x dollars and each gram makes y cups of coffee, what is the cost of one cup of coffee in terms of x and y?

Solution :

12 grams of coffee ----> x dollars

1 gram of coffee ----> ˣ⁄₁₂ dollars

y cups of coffee ----> 1 gram of coffee

y cups of coffee ----> ˣ⁄₁₂ dollars

1 cup of coffee ----> ˣ⁄₁₂y dollars

The cost of one cup of coffee is ˣ⁄₁₂y dollars.