Worksheet: Direct Variation

In this worksheet, we will practice describing direct variation between two variables and using this to solve word problems.

Q1:

The velocity, ๐‘ฃ, of an object in m/s is directly proportional to the time, ๐‘ก, in seconds. This relationship can be expressed in the equation ๐‘ฃ=9.8๐‘ก. Find the value of ๐‘ก when ๐‘ฃ=637 m/s.

Q2:

Given that ๐‘ฆ varies directly as ๐‘ฅ, write an equation for ๐‘ฆ in terms of ๐‘ฅ using ๐‘˜ as a non-zero constant.

  • A ๐‘ฆ = ๐‘˜ ๐‘ฅ
  • B ๐‘ฆ = ๐‘˜ ๐‘ฅ
  • C ๐‘ฆ = ๐‘˜ ๐‘ฅ ๏Šจ
  • D ๐‘ฆ = ๐‘˜ ๐‘ฅ ๏Šจ

Q3:

If ๐‘ฆโˆ๐‘ฅ and ๐‘ฆ=14 when ๐‘ฅ=6, determine the constant of proportionality.

  • A14
  • B 3 7
  • C84
  • D6
  • E 7 3

Q4:

A car moves with a uniform velocity where the distance varies directly with time. If the car travels a distance of 231 km in 9 hours, how far will it travel in 15 hours?

Q5:

An object that weighs 120 N on the Earth weighs 20 N on the moon. Given that the weight of an object on Earth is directly proportional to its weight on the moon, find the weight of an object on the moon given that its weight on Earth is 126 N.

Q6:

Given that ๐‘ฆ varies directly as ๐‘ฅ๏Šจ, write an equation for ๐‘ฆ in terms of ๐‘ฅ using ๐‘˜ as a non-zero constant.

  • A ๐‘ฆ = ๐‘˜ ๐‘ฅ ๏Šจ
  • B ๐‘ฆ = ๐‘˜ ๐‘ฅ
  • C ๐‘ฆ = ๐‘˜ ๐‘ฅ ๏Šจ
  • D ๐‘ฆ = ๐‘˜ ๐‘ฅ

Q7:

If ๐‘ฆโˆ๐‘ฅ and ๐‘ฅ=75 when ๐‘ฆ=25, find the value of ๐‘ฆ when ๐‘ฅ=30.

Q8:

Does ๐‘ฆ vary directly with ๐‘ฅ? If so, what is the constant of variation?

๐‘ฅ 2 4 6 8
๐‘ฆ 3 4 5 6
  • Ano
  • Byes, 112
  • Cyes, 1
  • Dyes, 56
  • Eyes, 34

Q9:

Determine whether the following statement is true or false: A directly proportional relationship can always be represented by a straight line graph passing through the origin.

  • Afalse
  • Btrue

Q10:

A gas station charges $3 per gallon for gasoline, and $6 for a car wash. Daniel never washes his car there because he thinks it is too expensive, but he often buys gasoline there. Is the total money he spends at the station directly proportional to the amount of gasoline he buys?

  • Ano
  • Byes

Q11:

Olivia uses 212 cups of flour to make 20 cookies. She makes a second batch using 334 cups of flour to make 30 cookies. Is the amount of flour she uses directly proportional to the number of cookies she bakes?

  • Ano
  • Byes

Q12:

Given that ๐‘ฅโˆ๐‘ฆ, and ๐‘ฅ=18 when ๐‘ฆ=5, find the value of ๐‘ฅ๐‘ฆ.

  • A 5 1 8
  • B 5 3 2 4
  • C 1 8 5
  • D 3 2 4 5

Q13:

Which of the following relations represents a direct variation between the two variables ๐‘ฅ and ๐‘ฆ?

  • A ๐‘ฆ = ๐‘ฅ + 2
  • B ๐‘ฅ 5 = ๐‘ฆ 4
  • C ๐‘ฅ 6 = 3 ๐‘ฆ
  • D ๐‘ฅ ๐‘ฆ = 6

Q14:

Which of the given graphs represents the direct variation between ๐‘ฅ and ๐‘ฆ?

  • A(b)
  • B(c)
  • C(a)
  • D(d)

Q15:

Given that ๐‘ฆ varies directly with ๐‘ฅ, and ๐‘ฆ=18 when ๐‘ฅ=6, write an equation for ๐‘ฆ in terms of ๐‘ฅ.

  • A ๐‘ฆ = ๐‘ฅ 6
  • B ๐‘ฆ = ๐‘ฅ + 1 2
  • C ๐‘ฆ = 3 ๐‘ฅ
  • D ๐‘ฆ = 6 ๐‘ฅ
  • E ๐‘ฆ = ๐‘ฅ 3

Q16:

A recipe for 4 people requires 440 g of spaghetti. The quantity of spaghetti is proportional to the number of people. What is the corresponding constant of proportionality (unit rate)?

  • A110 g/person
  • B440 g/person
  • C1,760 persons/g
  • D1,760 g/person
  • E110 persons/g

Q17:

The total cost for pizzas is proportional to the number of pizzas, as shown in the graph.

Find the unit rate.

  • A$60 per pizza
  • B30ยข per pizza
  • C15ยข per pizza
  • D$15 per pizza
  • E$30 per pizza

Q18:

The amount of meat required to feed a captive lion is given by the equation ๐‘ค=9๐‘‘, where ๐‘ค is the weight of the meat in kilograms needed to feed a lion for ๐‘‘ days. What is the unit rate of this proportional relationship?

Q19:

The amount of caffeine in a cup of coffee is roughly 95 mg. What is the constant of proportionality, or unit rate, between the amount of caffeine ingested and the number of cups taken?

Q20:

Points on a line are given in the table. Does ๐‘ฆ vary directly with ๐‘ฅ? Is so, what is the constant of variation?

๐‘ฅ 3 6 9 12
๐‘ฆ 2 4 6 8
  • Ayes, 112
  • Byes, 23
  • Cno

Q21:

The length of a rectangle is 8 m and its width is 5 m. What is the perimeter of a similar rectangle with length 19 meters?

  • A 3 0 7 8 m
  • B 1 1 7 8 m
  • C 9 8 4 5 m
  • D 4 9 2 5 m
  • E 6 1 3 4 m

Q22:

If ๐‘ฅ=4๐‘ฆโˆ’84, then .

  • A ๐‘ฅ โˆ 4 ๐‘ฆ
  • B ๐‘ฅ โˆ ๐‘ฆ
  • C ๐‘ฅ โˆ 4 ๐‘ฆ + 8 4
  • D ๐‘ฅ โˆ ๐‘ฆ โˆ’ 2 1

Q23:

The cost of hiring a small car for 4 days is $170. What would the cost to hire the car for a week at the same rate be? Give your answer to two decimal places if necessary.

Q24:

The instructions on a packet of fertilizer state that 500 g will treat an area of 60 square meters if mixed correctly. How much fertilizer is needed to treat a garden whose area is 100 square meters? Give your answer to the nearest gram if necessary.

Q25:

A website is selling e-books. It offers a discount of $0.50 for every referral you make. Is the amount of money saved by that offer directly proportional to the number of referrals you make?

  • Ayes
  • Bno

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