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, 𝑣 , of an object in m/s is directly proportional to the time, 𝑑 , in seconds. This relationship can be expressed in the equation 𝑣 = 9 . 8 𝑑 . Find the value of 𝑑 when 𝑣 = 6 3 7 m/s.

Q2:

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

  • A 𝑦 = π‘˜ π‘₯ 2
  • B 𝑦 = π‘˜ π‘₯
  • C 𝑦 = π‘˜ π‘₯ 2
  • D 𝑦 = π‘˜ π‘₯

Q3:

If 𝑦 ∝ π‘₯ and 𝑦 = 1 4 when π‘₯ = 6 , determine the constant of proportionality.

  • A 14
  • B 3 7
  • C 6
  • D 7 3
  • E 84

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 𝑦 varies directly as π‘₯ 2 , 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‘₯ 2
  • D 𝑦 = π‘˜ π‘₯ 2

Q7:

If 𝑦 ∝ π‘₯ and π‘₯ = 7 5 when 𝑦 = 2 5 , find the value of 𝑦 when π‘₯ = 3 0 .

Q8:

Does 𝑦 vary directly with π‘₯ ? If so, what is the constant of variation?

π‘₯ 2 4 6 8
𝑦 3 4 5 6
  • A yes, 1 1 2
  • B yes, 3 4
  • C yes, 1
  • Dno
  • E yes, 5 6

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.

  • Atrue
  • Bfalse

Q10:

A gas station charges $3 per gallon for gasoline, and $6 for a car wash. Adam 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?

  • Ayes
  • Bno

Q11:

Dalia uses 2 1 2 cups of flour to make 20 biscuits. She makes a second batch using 3 3 4 cups of flour to make 30 biscuits. Is the amount of flour she uses directly proportional to the number of biscuits she bakes?

  • Ayes
  • Bno

Q12:

Given that π‘₯ ∝ 𝑦 , and π‘₯ = 1 8 when 𝑦 = 5 , find the value of π‘₯ 𝑦 .

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

Q13:

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

  • A 𝑦 = π‘₯ + 2
  • B π‘₯ 6 = 3 𝑦
  • C π‘₯ 𝑦 = 6
  • D π‘₯ 5 = 𝑦 4

Q14:

Which of the given graphs represents the direct variation between π‘₯ and 𝑦 ?

  • A(c)
  • B(a)
  • C(d)
  • D(b)

Q15:

Given that 𝑦 varies directly with π‘₯ , and 𝑦 = 1 8 when π‘₯ = 6 , write an equation for 𝑦 in terms of π‘₯ .

  • A 𝑦 = 6 π‘₯
  • B 𝑦 = π‘₯ 3
  • C 𝑦 = π‘₯ 6
  • D 𝑦 = 3 π‘₯
  • E 𝑦 = π‘₯ + 1 2

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)?

Q17:

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

Find the unit rate.

  • A15Β’ per pizza
  • B$30 per pizza
  • C30Β’ per pizza
  • D$15 per pizza
  • E$60 per pizza

Q18:

The amount of meat required to feed a captive lion is given by the equation 𝑀 = 9 𝑑 , where 𝑀 is the weight of the meat in kilograms needed to feed a lion for 𝑑 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 𝑦 vary directly with π‘₯ ? Is so, what is the constant of variation?

π‘₯ 3 6 9 12
𝑦 2 4 6 8
  • A no
  • B yes, 1 1 2
  • C yes, 2 3

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 metres?

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

Q22:

Engy decided to make pizzas. The graph shows the number of tomatoes needed for a given number of pizzas.

Determine which of the following statements is true.

  • AThe point ( 5 , 1 0 ) shows that she needs 5 tomatoes to make 10 pizzas.
  • BThe point (6, 3) shows that to make 6 pizzas she needs 3 tomatoes.
  • CThe point ( 1 0 , 5 ) shows that she needs 5 tomatoes to make 10 pizzas.
  • DThe point (3, 6) shows that to make 3 pizzas she needs 6 tomatoes.
  • EThe point (8, 4) shows that she needs 8 tomatoes to make 4 pizzas.

Q23:

If π‘₯ = 4 𝑦 βˆ’ 8 4 , then .

  • A π‘₯ ∝ 4 𝑦 + 8 4
  • B π‘₯ ∝ 𝑦
  • C π‘₯ ∝ 4 𝑦
  • D π‘₯ ∝ 𝑦 βˆ’ 2 1

Q24:

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.

Q25:

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

  • A 600 g
  • B 300 g
  • C 450 g
  • D 833 g
  • E 124 g

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