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:

𝑦 varies inversely with π‘₯. The constant of proportionality is 6. Which of the following equations represents this relation?

  • A 𝑦 = 6 π‘₯
  • B 𝑦 = 6 π‘₯ 
  • C 𝑦 = 6 π‘₯ 
  • D 𝑦 = 6 βˆ’ π‘₯ 

Q3:

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

π‘₯ 2 4 70
𝑦 70 35 2
  • A420
  • B 3 1 4 0
  • C 1 5 5 9
  • D 4 6 2 3

Q4:

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

  • A 𝑦 = π‘˜ π‘₯ 
  • B 𝑦 = π‘˜ π‘₯
  • C 𝑦 = π‘˜ π‘₯
  • D 𝑦 = π‘˜ π‘₯ 

Q5:

If π‘¦βˆ1√π‘₯ then which of the following is true?

  • A π‘₯ is inversely proportional to 𝑦
  • B π‘₯ is directly proportional to π‘¦οŠ¨
  • C π‘₯ is inversely proportional to π‘¦οŠ¨
  • D π‘₯ is inversely proportional to π‘¦οŠ©
  • E π‘₯ is directly proportional to 𝑦

Q6:

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.

Q7:

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

π‘₯ 2 7 28
𝑦 28 8 2

Q8:

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

  • A 𝑦 = π‘˜ π‘₯ 
  • B 𝑦 = π‘˜ π‘₯ 
  • C 𝑦 = π‘˜ π‘₯
  • D 𝑦 = π‘˜ π‘₯

Q9:

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

Q10:

The weight of an object above Earth’s surface varies inversely with the square of the distance from Earth’s center. If a body weighs 50 pounds when it is 3,960 miles from the earth’s center, what would it weigh if it was 3,970 miles from Earth’s center?

Q11:

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

  • A π‘₯ 
  • B 1 π‘₯
  • C 1 π‘₯ 
  • D π‘₯

Q12:

𝑦 varies inversely with π‘₯. The constant of proportionality is 15. Which of the following equations represents this relation?

  • A 𝑦 = 1 5 π‘₯
  • B 𝑦 = 1 5 π‘₯ 
  • C 𝑦 = 1 5 π‘₯ 
  • D 𝑦 = 1 5 βˆ’ π‘₯ 

Q13:

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

Q14:

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.

Q15:

The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1,200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

Q16:

The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

Q17:

Scarlett 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.

  • A 𝑠 = 𝑛 5 , 0 0 0
  • B 𝑠 = 5 0 0 𝑛
  • C 𝑠 = 5 0 𝑛
  • D 𝑠 = 5 , 0 0 0 𝑛
  • E 𝑠 = 𝑛 5

Q18:

If 𝑦=π‘Žβˆ’8, π‘¦βˆ1π‘₯, and π‘Ž=20 when π‘₯=8, find 𝑦 when π‘₯=6.

  • A 1 1 6
  • B 6 4 3
  • C 3 6 4
  • D16

Q19:

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

  • A π‘Ž ∝ 𝑏 
  • B π‘Ž ∝ 1 𝑏 
  • C π‘Ž ∝ 1 𝑏
  • D π‘Ž ∝ 𝑏

Q20:

Given that 𝑦=10+𝑏, where 𝑏 varies inversely with π‘₯, and 𝑦=130 when π‘₯=12, determine the value of 𝑦 when π‘₯=βˆ’10.

  • A 7 4 5
  • B 1 0 3 1 0
  • C 1 0 1 0 3
  • D 5 7 4

Q21:

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.

  • A π‘š = 1 0 0 𝑛
  • B π‘š = 1 , 0 0 0 𝑛
  • C π‘š = 1 0 𝑛
  • D π‘š = 𝑛 1 , 0 0 0
  • E π‘š = 𝑛 1 0 0

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

Q22:

If 𝑦=π‘Žβˆ’6, π‘¦βˆ1π‘₯, and π‘Ž=10 when π‘₯=5, which of these equations represents the relation between π‘₯ and 𝑦.

  • A 𝑦 = 1 0 0 π‘₯ βˆ’ 6 
  • B 𝑦 = 1 π‘₯ βˆ’ 6 
  • C 𝑦 = 2 0 π‘₯ 
  • D 𝑦 = 6 βˆ’ 1 0 0 π‘₯ 
  • E 𝑦 = 1 0 0 π‘₯ 

Q23:

The height of a right circular cylinder β„Ž varies inversely with the square of its radius π‘Ÿ. If β„Ž=93cm when π‘Ÿ=7.5cm, determine β„Ž when π‘Ÿ=1.5cm.

Q24:

Given that π‘₯∝1βˆšπ‘¦οŠ¨οŽ’, and π‘₯=3 when 𝑦=1,728, find 𝑦 when π‘₯=6.

Q25:

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?

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