RATIO RATES AND PROPORTIONS WORKSHEET

Question 1 :

The density of an object is equal to the mass of the object divided by the volume of the object. Find the mass of an object in grams with a volume of 0.01 m³ and a density of 4.54 grams per cubic centimeters? (1 m = 100 cm)

Question 2 :

Jason and Donny painted a house and received $1,200. To complete the painting job Jason painted 4 hours 25 minutes and Donny spent 2 hours and 15 minutes. If they split the $1,200 in proportion to the amount of time each spent painting, how much did Donny receive?

Question 3 :

The tennis balls in a bag are either white or yellow. If the ratio of white balls to yellow balls is ³⁄₁₀, which of the following could not be the number of balls in the bag?

(A)  26

(B)  39

(C)  42

(D)  52

Question 4 :

A car is traveling at a constant rate of x miles per hour. How many miles will the car travel in y minutes?

Question 5 :

A tree is 8 feet tall and grows 8 inches each year. In how many years will the tree reach a height of 30 feet?

Question 6 :

Aaron reads x pages of a science fiction book in m minutes. If he continues reading at this rate, what will be the number of pages he reads in 20m seconds?

Question 7 :

If ˣ⁄y = 1, what is the value of x - y - 1?

Question 8 :

In a certain room the ratio of males to females is 4 to 5. After 8 males enter the room, the ratio of males to females is 6 to 5. What is the total number of people in the room before the additional males enter the room?

Question 9 :

A person is born every 5 seconds and a person dies every 12 seconds. How many seconds does it take for the population to grow by one person?

Question 10 :

Steve is going to paint a wall that measures 9 feet by 12 feet. If one gallon of paint is needed for each s square foot of wall and each gallon costs g dollars, in terms of s and g how much does it cost to paint the entire wall?

(A)  ¹⁰⁸⁄gs

(B)  ᵍˢ⁄₁₀₈

(B)  ¹⁰⁸ˢ⁄g

(B)  ¹⁰⁸ᵍ⁄s

Question 11 :

If 2 inches are equivalent to 5 centimeters, how many square centimeters are in one square inch?

Question 12 :

A large painting has a length of 18 inches and a width of 12 inches. If each dimension is reduced by x inches to make the ratio of length to width 5 to 3, what is the value of x?

Answers

1. Answer :

1 m = 100 cm

(1 m)³ = (100 cm)³

1 m³ = 1,000,000 cm³

0..01 ⋅ 1 m³ = 0.01 ⋅ 1,000,000 cm³

0..01 m³ = 10,000 cm³

Mass = (10000 cm³)(4.54 grams/cm³)

Mass = 45,400 grams

2. Answer :

Time spent by Jason :

= 4 hours 25 minutes

= 4²⁵⁄₆₀ hours

= 4⁵⁄₁₂ hours

Time spent by Donny :

= 2 hours 15 minutes

= 2¹⁵⁄₆₀ hours

= 2¼ hours

Ratio of time spent by Jason and Donny :

= 4⁵⁄₁₂: 2¼

= ⁵³⁄₁₂ : ⁹⁄₄

= 12(⁵³⁄₁₂) : 12( ⁹⁄₄)

= 53 : 27

Jason and Donny have to share the total money in the ratio 53 : 27.

Jason's share = 53x

Donny's share = 27x

Given : Total money received $1,200.

53x + 27x = 1200

80x = 1200

x = 15

Money received by Donny :

= 27(15)

= $405

3. Answer :

Given : The ratio of white balls to yellow balls is ³⁄₁₀.

white : yellow = 3 : 10

number of white balls = 3x

number of yellow balls = 10x

Total number of balls in the bag :

= 3x + 10x

= 13x

So, total number of balls in the bag is a multiple of 13.

In the given choices, 42 is not a multiple of 13.

Therefore, the correct answer is option (C).

4. Answer :

Given : The car is traveling at a constant rate of x miles per hour.

1 hour ----> x miles

60 minutes ----> x miles

y(1 minute) ----> y(ˣ⁄₆₀) miles

y minutes ----> ˣʸ⁄₆₀ miles

The the car will travel ˣʸ⁄₆₀ miles in yminutes.

5. Answer :

Let x be the number of years required for the tree to reach a height of 30 feet.

Given : The tree grows 8 inches each year.

8 inches = ⁸⁄₁₂ or ⅔ feet

The tree grows ⅔ feet each year.

Present height + Height grow in x years = 30

8 ft. +  (⅔)x ft. = 30 ft.

8 +  (⅔)x = 30

Multiply both sides by 3.

24 + 2x = 90

Subtract 24 from both sides.

2x = 66

Divide both sides by 2.

x = 33

It will take 33 years for the tree to reach a height of 30 feet.

6. Answer :

Aaron reads 18 pages of a science fiction book in m minutes. If he continues reading at this rate, what will be the number of pages he reads in 20m seconds?

m minutes = 18 pages

1 minute = ¹⁸⁄m pages

60 seconds = ¹⁸⁄m pages

1 second = ¹⁸⁄₆₀m pages

20m(1 second) = 20m(¹⁸⁄₆₀m pages) 

20m seconds = 3 pages

Aaron will read 3 pages in 20m seconds.

7. Answer :

ˣ⁄y = 1

Multiply both sides by y.

x = y

Subtract y from both sides.

x - y = 1

Subtract 1 from both sides.

x - y - 1 = 0

8. Answer :

Let m and f be the original number of males and females respectively.

Given : The ratio of males to females is 4 to 5.

m ; f = 4 : 5

ᵐ⁄f = ⅘

5m = 4f ----(1)

Given : After 8 males enter the room, the ratio of males to females is 6 to 5.

(m + 8) : f = 6 : 5

⁽ᵐ ⁺ ⁸⁾⁄f = ⁶⁄₅

5(m + 8) = 6f

5m + 40 = 6f

Substitute 5m = 4f.

4f + 40 = 6f

40 = 2f

20 = f

Substitute f = 20 in (1).

5m = 4(20)

5m = 80

m = 16

The total number of people in the room before the additional males enter the room :

= 16 + 20

= 36

9. Answer :

Every 5 seconds, a person is born :

5 seconds ----> 1 person

12(5 seconds) ----> 12(1 person)

60 seconds ----> 12 people

1 minute ----> 12 people

Every 12 seconds, a person dies :

12 seconds ----> 1 person

5(12 seconds) ----> 5(1 person)

60 seconds ----> 5 people

1 minute ----> 5 people

Every minute, 12 people are born and 5 people die.

Every minute, the population grows by (12 - 5) or 7 people.

7 people ----> 1 minute

7 people ----> 60 seconds

1 person ----> ⁶⁰⁄₇ seconds

1 person ----> 8⁴⁄₇ seconds

It takes 8⁴⁄₇ seconds for the population to grow by one person.

10. Answer :

Total area of the wall :

= 9 ⋅ 12

= 108 ft²

Given : One gallon of paint is needed for each s square foot of wall.

Let x be the number of gallons of paint needed to paint 108 ft².

1 : s = x : 108

¹⁄s = ˣ⁄₁₀₈

x = ¹⁰⁸⁄s gallons

¹⁰⁸⁄s gallons of paint needed to paint the entire wall with the area 108 ft².

Given : The cost of 1 gallon of paint is g dollars.

Now, we have to find the cost of ¹⁰⁸⁄s gallons of paint.

1 gallon ----> g dollars

(¹⁰⁸⁄s)(1 gallon) ----> (¹⁰⁸⁄s)(g dollars)

¹⁰⁸⁄s gallons ----> ¹⁰⁸ᵍ⁄s dollars

It costs ¹⁰⁸ᵍ⁄s dollars to paint the entire wall.

Therefore, the correct answer is option (D).

11. Answer :

Given : 2 inches are equivalent to 5 centimeters.

2 in = 5 cm

(2 in)² = (5 cm)²

2² in² = 5² cm²

4 in² = 25 cm²

1 in² = ²⁵⁄₄ cm²

There are ²⁵⁄₄ square centimeters in 1 square inch.

12. Answer :

From the given information,

(18 - x) : (12 - x) = 5 : 3

⁽¹⁸ ⁻ ˣ⁾⁄₍₁₂ ₋ ₓ₎ = ⁵⁄₃

3(18 - x) = 5(12 - x)

54 - 3x = 60 - 5x

2x = 6

x = 3

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