Worksheet: Inverse Variation

In this worksheet, we will practice creating formulas linking two quantities that vary directly and indirectly.

Q1:

𝑦 varies inversely with π‘₯. Given that 𝑦=8 when π‘₯=7, what is the constant of proportionality?

Q2:

Decide if π‘₯ varies directly or inversely with 𝑦 and use this to find the value of 𝑦 when π‘₯=3.

π‘₯2470
𝑦70352
  • A420
  • B3140
  • C1559
  • D4623

Q3:

For a rectangle of fixed area, the length 𝑙 varies inversely with its width 𝑀. Given that 𝑙=22cm when 𝑀=16cm, determine the value of 𝑙 when 𝑀=44cm.

Q4:

The table below shows how 𝑦 varies with π‘₯. Use this information to find the value of π‘₯ when 𝑦=14.

π‘₯2728
𝑦2882

Q5:

Given that 𝑦 varies inversely as π‘₯, write an equation for 𝑦 in terms of π‘₯ using π‘˜ as a non-zero constant.

  • A𝑦=π‘˜π‘₯
  • B𝑦=π‘˜π‘₯
  • C𝑦=π‘˜π‘₯
  • D𝑦=π‘˜π‘₯

Q6:

At the beginning of the month, Jacob bought 70 eggs. Every morning he eats 2 eggs for breakfast. Is the number of remaining eggs proportional to the number of the days that have passed?

  • Ayes
  • Bno

Q7:

If 𝑧=π‘šπ‘₯, where π‘š is a constant, then π‘§βˆ.

  • Aπ‘₯
  • B1π‘₯
  • C1π‘₯
  • Dπ‘₯

Q8:

𝑦 varies inversely with π‘₯. Given that 𝑦=8 when π‘₯=12, what is the constant of proportionality?

Q9:

For a rectangle of fixed area, the length 𝑙 varies inversely with its width 𝑀. Given that 𝑙=13cm when 𝑀=11cm, determine the value of 𝑙 when 𝑀=13cm.

Q10:

Sophia buys 5 liters of soda to share between the children at a party. The amount of soda each child will receive varies inversely with the number of children at the party. Write an equation for 𝑠, the amount of soda in milliliters each child will receive, in terms of 𝑛, the number of children at the party.

  • A𝑠=𝑛5,000
  • B𝑠=500𝑛
  • C𝑠=50𝑛
  • D𝑠=5,000𝑛
  • E𝑠=𝑛5

Q11:

Determine the correct relation from the following choices, given that π‘Žπ‘βˆ’40π‘Žπ‘=βˆ’400οŠͺ, π‘Žβˆˆβ„, and π‘βˆˆβ„.

  • Aπ‘Žβˆπ‘οŠ¨
  • Bπ‘Žβˆ1π‘οŠ¨
  • Cπ‘Žβˆ1𝑏
  • Dπ‘Žβˆπ‘

Q12:

A group of scouts receives a donation of $1,000 to fund places on an international jamboree. The amount each scout receives for their trip varies inversely with the number of scouts from the group going to the jamboree.

Write an equation for π‘š, the amount each scout receives, in terms of 𝑛, the number of scouts from the group who are going to the jamboree.

  • Aπ‘š=100𝑛
  • Bπ‘š=1,000𝑛
  • Cπ‘š=10𝑛
  • Dπ‘š=𝑛1,000
  • Eπ‘š=𝑛100

If 25 scouts from the group are going to the jamboree, how much will each scout receive from the donation?

Q13:

The number of hours 𝑛 needed for carrying out a certain task varies inversely with the number of workers who carry out the task. If the task is carried out by 23 workers in 35 hours, what is the time needed for 115 workers to carry out the task?

Q14:

A copy center is printing leaflets for a local school. The time it takes for the job to be completed varies inversely with the number of copiers used, and if 4 copiers are used, the job will take 34 of an hour.

How many copiers are required to complete the job in half an hour?

Letting 𝑑 be the time in hours to complete the job using 𝑛 copiers, what is the value of the product, 𝑑𝑛?

Write an equation for 𝑑 in terms of 𝑛.

  • A𝑑=12𝑛
  • B𝑑=12𝑛
  • C𝑑=3𝑛
  • D𝑑=3𝑛
  • E𝑑=𝑛3

Q15:

If π‘¦βˆ1π‘₯, and π‘₯=78 when 𝑦=57, find the value of 𝑦 when π‘₯=6.

Q16:

Which of the following relations represents an inverse variation between the two variables π‘₯ and 𝑦?

  • A𝑦=π‘₯+3
  • B𝑦=6π‘₯
  • Cπ‘₯𝑦=72
  • Dπ‘₯𝑦=14

Q17:

Community groups have been invited to apply for a grant from a fund. The amount a group will receive varies inversely with the number of successful applications.

If there are 45 successful applications, each group will receive $800.

How much will each group receive if there are 360 successful applications?

How many successful applications mean that each group receives $3,600?

How much will each group receive if there are 150 successful applications?

What is the total value of the fund?

Write an equation for π‘š, the number of dollars each group receives when 𝑛 groups make successful applications.

  • Aπ‘š=36,000𝑛
  • Bπ‘š=𝑛3,600
  • Cπ‘š=360,000𝑛
  • Dπ‘š=3,600𝑛
  • Eπ‘š=𝑛36,000

Q18:

A building contractor is constructing part of the perimeter wall for a school. The wall will be the same height all along its length. The length of time it will take to build the wall varies inversely with the number of bricklayers. A team of 12 bricklayers is expected to take 30 days to build the wall.

Write an equation for the number of days, 𝑑, it would take a team of 𝑏 bricklayers to build the wall.

  • A𝑑=360𝑏
  • B𝑑=30𝑏
  • C𝑑=30𝑏
  • D𝑑=12𝑏
  • E𝑑=360𝑏

How many bricklayers are required if the wall is to be finished in 20 days?

Q19:

Given that π‘Ž varies inversely with 𝑏, and π‘Ž=5 when 𝑏=34, write an equation for π‘Ž in terms of 𝑏.

  • Aπ‘Ž=15𝑏
  • Bπ‘Ž=203𝑏
  • Cπ‘Ž=154𝑏
  • Dπ‘Ž=𝑏15
  • Eπ‘Ž=4𝑏15

Q20:

The time taken to complete a car journey at a steady speed varies inversely with the speed of the car.

Most of the quickest route from William’s Corner to Round Rock is on the I-35. At 45 mph, this part of the journey takes 2 hours and 40 minutes.

How long would the journey take at 75 mph?

  • A1 hour and 15 minutes
  • B1 hour and 36 minutes
  • C1 hour and 55 minutes
  • D2 hours and 15 minutes
  • E2 hours and 36 minutes

Write an equation for 𝑑, the time it takes to cover this part of the journey at speed 𝑠.

  • A𝑑=60𝑠
  • B𝑑=120𝑠
  • C𝑑=𝑠120
  • D𝑑=𝑠60
  • E𝑑=45𝑠

How much of the route is on the I-35?

Q21:

A company making footballs decided to cut the selling price of each football by 30%. Assuming all their footballs are identical and the same price, if they wanted to get the same amount of money from sales, by what percentage will they need to increase the number of footballs sold? Round your answer to the nearest whole number.

Q22:

𝑧 varies inversely with 𝑦 such that π‘₯=𝑧+3. Given that 𝑧=6 when 𝑦=4, then find the value of 𝑦 when π‘₯=6.

Q23:

Suppose that 𝑦 is inversely proportional to π‘₯ and that 𝑦=βˆ’4 when π‘₯=βˆ’30. Write the equation, giving 𝑦 in terms of π‘₯.

  • A𝑦=120π‘₯
  • B𝑦=7.5π‘₯
  • C𝑦=60π‘₯
  • D𝑦=4π‘₯30
  • E𝑦=7.5π‘₯

Q24:

Given that π‘₯π‘¦βˆ’10π‘₯𝑦+25=0, which of the following describes the type of proportionality between 𝑦 and π‘₯.

  • Aπ‘¦βˆπ‘₯
  • Bπ‘¦βˆ1π‘₯
  • Cπ‘¦βˆ1π‘₯
  • Dπ‘¦βˆπ‘₯

Q25:

Which of the following represents the relation between π‘₯ and 𝑦?

π‘₯445599110
𝑦6075135150
  • Adirect variation
  • Binverse variation
  • Csquare of multiplicative inverse
  • Ddoes not represent direct variation or inverse variation

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.