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Worksheet: Direct Variation and Its Applications

Q1:

A car moves at a uniform velocity and travels 240 km in 3 hours. If the car continues at the same velocity, what distance will be covered in 5 hours?

Q2:

Matthew works 37 hours a week and is paid $416.25. Given that 𝑀 , the amount in dollars that Matthew earns, varies directly with β„Ž , the number of hours he works, write an equation for 𝑀 in terms of β„Ž .

  • A 𝑀 = 1 2 . 2 5 β„Ž
  • B 𝑀 = 1 0 . 2 5 β„Ž
  • C 𝑀 = 9 . 4 6 β„Ž
  • D 𝑀 = 1 1 . 2 5 β„Ž
  • E 𝑀 = 3 . 1 8 β„Ž

Q3:

The amount a babysitter earns varies directly with the number of hours they work. A babysitter earns $25.50 for 3 hours of babysitting.

How much would they earn for 2 hours of babysitting?

Write an equation for 𝑀 , the number of dollars earned for 𝑑 hours of babysitting.

  • A 𝑀 = 8 . 5 𝑑
  • B 𝑀 = 𝑑 8 . 5
  • C 𝑀 = 1 7 𝑑
  • D 𝑀 = 4 . 2 5 𝑑
  • E 𝑀 = 𝑑 4 . 2 5

Q4:

If π‘₯ varies directly with 𝑦 and π‘₯ = 1 5 when 𝑦 = 2 5 , what is the value of π‘₯ when 𝑦 = 5 ?

Q5:

If π‘Ž varies directly with 𝑏 and π‘Ž = 2 4 when 𝑏 = 8 4 , find the value of 𝑏 when π‘Ž = 1 0 .

Q6:

Liam is 1.6 m tall, and his shadow is 4 m long. He is standing near a tree whose shadow is 22 m long.

How tall is the tree?

Q7:

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.

Q8:

Given that π‘Ž varies directly with 𝑏 , and π‘Ž = 1 5 when 𝑏 = 1 0 , write an equation for π‘Ž in terms of 𝑏 .

  • A π‘Ž = 𝑏 + 5
  • B π‘Ž = 2 𝑏 3
  • C π‘Ž = 2 𝑏 βˆ’ 5
  • D π‘Ž = 3 𝑏 2
  • E π‘Ž = 𝑏 2

Q9:

Given that 𝑦 varies directly with π‘₯ , and 𝑦 = 4 when π‘₯ = 2 0 , write an equation for 𝑦 in terms of π‘₯ .

  • A 𝑦 = 4 π‘₯
  • B 𝑦 = 5 π‘₯
  • C 𝑦 = π‘₯ 4
  • D 𝑦 = π‘₯ 5
  • E 𝑦 = π‘₯ βˆ’ 1 6

Q10:

Given that π‘Ž varies directly with 𝑏 , and π‘Ž = 3 5 when 𝑏 = 4 2 , write an equation for π‘Ž in terms of 𝑏 .

  • A π‘Ž = 𝑏 βˆ’ 7
  • B π‘Ž = 6 𝑏 5
  • C π‘Ž = 5 𝑏 7
  • D π‘Ž = 5 𝑏 6
  • E π‘Ž = 𝑏 6

Q11:

Use the data in the table to find the values of π‘Ž and 𝑏 .

π‘₯ 136 204 𝑏 884 1 360
𝑦 70 π‘Ž 420 455 700
  • A π‘Ž = 3 5 , 𝑏 = 6 8
  • B π‘Ž = 3 , 𝑏 = 1 2
  • C π‘Ž = 1 1 , 𝑏 = 5
  • D π‘Ž = 1 0 5 , 𝑏 = 8 1 6
  • E 𝑏 = 1 0 5 , π‘Ž = 8 1 6

Q12:

Which table does NOT show π‘₯ varying directly with 𝑦 ?

  • A
    π‘₯ 10 20 30
    𝑦 2 4 6
  • B
    π‘₯ 1 2 3
    𝑦 12 24 36
  • C
    π‘₯ 2 0 βˆ’ 2
    𝑦 8 0 βˆ’ 8
  • D
    π‘₯ 5 3 1
    𝑦 6 10 30
  • E
    π‘₯ 2 4 6
    𝑦 1.5 3 4.5

Q13:

𝑦 = π‘Ž + 𝑏 , where π‘Ž is a constant and 𝑏 varies directly with π‘₯ . Given that 𝑦 = 9 when π‘₯ = 2 , and 𝑦 = 4 when π‘₯ = 1 , what is the value of 𝑦 when π‘₯ = 9 .

Q14:

Given that 𝑦 ∝ ο€Ό 1 π‘₯  t h e m u l t i p l i c a t i v e i n v e r s e o f 2 , and 𝑦 = 4 4 when π‘₯ = 2 , then find the value of 𝑦 when π‘₯ = 4 .

Q15:

Jennifer 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.

Q16:

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.

Q17:

π‘₯ and 𝑦 are two variables, where 𝑦 varies directly with the multiplicative inverse of ο€Ό 1 π‘₯  3 . If 𝑦 = 3 5 when π‘₯ = 1 0 , determine the values of 𝑦 when π‘₯ ∈ { 3 , 4 , 5 } .

  • A 𝑦 = 3 5 2 7 when π‘₯ = 3 , 𝑦 = 3 5 6 4 when π‘₯ = 4 , 𝑦 = 7 2 5 when π‘₯ = 5
  • B 𝑦 = 1 0 2 7 when π‘₯ = 3 , 𝑦 = 5 3 2 when π‘₯ = 4 , 𝑦 = 2 2 5 when π‘₯ = 5
  • C 𝑦 = 1 8 9 2 0 when π‘₯ = 3 , 𝑦 = 1 1 2 5 when π‘₯ = 4 , 𝑦 = 1 7 5 4 when π‘₯ = 5
  • D 𝑦 = 1 8 9 2 0 0 when π‘₯ = 3 , 𝑦 = 5 6 2 5 when π‘₯ = 4 , 𝑦 = 3 5 8 when π‘₯ = 5

Q18:

Given that 𝑦 ∝ π‘₯ 2 3 , where 𝑦 = βˆ’ 1 5 when π‘₯ = 7 , find the relation between π‘₯ and 𝑦 .

  • A 𝑦 = βˆ’ 2 2 5 3 4 3 π‘₯ 2 3
  • B 𝑦 = 3 4 3 2 2 5 π‘₯ 2 3
  • C 𝑦 = βˆ’ 3 4 3 2 2 5 π‘₯ 2 3
  • D 𝑦 = 2 2 5 3 4 3 π‘₯ 2 3

Q19:

Which of the following points represents the unit rate in the graph?

  • A ( 2 , 2 )
  • B ( 2 , 1 )
  • C ( 1 , 1 )
  • D ( 1 , 2 )
  • E ( 2 , 5 )

Q20:

A fixed menu at a restaurant costs $25 for two people. A group of 10 friends went to the restaurant, and everyone ordered the fixed menu.

How much was the check?

Q21:

Given that ο€Ή 3 8 π‘₯ + 9 𝑦  ∝ ο€Ή π‘₯ + 4 6 𝑦  2 2 , π‘₯ ∈ ℝ , and 𝑦 ∈ ℝ . Which of the following correctly expresses the relation.

  • A π‘₯ ∝ 1 𝑦
  • B π‘₯ ∝ 𝑦
  • C π‘₯ ∝ 1 𝑦 2
  • D π‘₯ ∝ 𝑦 2

Q22:

Scarlett prepared some mortar to repoint the joints of her garden brick wall. She filled 2 3 of a small plastic tray with cement, put it in a large bucket, and then added 2 3 4 trays of sand. Finally, she added water and mixed it all. How many trays of sand will she have to use if she wants to make mortar in the same proportions with one tray of cement?

  • A 6 5 6
  • B 3 5 1 2
  • C 6 1 6
  • D 4 1 8
  • E 5 1 2

Q23:

The graph shows how many tablespoons of sugar are needed to make a given number of glasses of lemonade.

Determine which of the following statements is NOT true.

  • A The unit rate of this proportional relationship is 3 tablespoons of sugar per glass of lemonade.
  • B The point ( 3 , 9 ) shows that 9 tablespoons of sugar are needed to make 3 glasses of lemonade.
  • C Any point ( π‘₯ , 𝑦 ) on this graph means that 𝑦 tablespoons of sugar are needed to make π‘₯ glasses of lemonade.
  • DThe recipe always requires the same number of tablespoons of sugar as that of lemonade glasses.
  • ETo make 20 glasses of lemonade, 60 tablespoons of sugar are needed.

Q24:

When making concrete, 20 kg of cement is used with 8 liters of water.

How much cement would be needed with 10 liters of water?

Q25:

Elizabeth and Chloe like making necklaces. The graphs below show how many necklaces they make against time. The red line belongs to Elizabeth and the blue one to Chloe.

Which of the following statements is not true?

  • AThe point (1, 40) shows that the unit rate for Denise is 40 necklaces per hour.
  • BThe point (1, 30) shows that the unit rate for Ann is 30 necklaces per hour.
  • CWe know that Denise is faster than Ann making necklaces because the blue line is above the red line.
  • DBoth girls make the same number of necklaces at the same time because they started at (0, 0).
  • EA higher unit rate gives a steeper line.