Lesson Worksheet: Joint Variation Mathematics • 9th Grade

In this worksheet, we will practice writing an equation to describe joint variation and using proportions to find other sets of values to solve real-world problems.


𝑦 varies jointly as π‘₯ and 𝑧. If 𝑦=15 when π‘₯=9 and 𝑧=8, find 𝑦 when π‘₯=16 and 𝑧=9.


𝑦 varies jointly as π‘₯ and the cube root of 𝑧. If 𝑦=120 when π‘₯=5 and 𝑧=8, find 𝑦 when π‘₯=10 and 𝑧=64.


𝑦 varies jointly as π‘₯ and the square of 𝑧. If 𝑦=18 when π‘₯=2 and 𝑧=3, find 𝑦 when π‘₯=6 and 𝑧=7.


Given that 𝑧 varies jointly with π‘₯ and 𝑦, such that 𝑧=12π‘₯𝑦, find π‘₯ when 𝑦=1 and 𝑧=2.

  • Aπ‘₯=14
  • Bπ‘₯=1
  • Cπ‘₯=2
  • Dπ‘₯=12
  • Eπ‘₯=4


A quantity π‘₯ varies directly with 𝑦 and inversely with 𝑧. Given that π‘˜ is the constant of variation, find the equation that models π‘₯.

  • Aπ‘₯=𝑧𝑦
  • Bπ‘₯=𝑦𝑧
  • Cπ‘₯=π‘˜π‘§π‘¦
  • Dπ‘₯=π‘˜π‘¦π‘§
  • Eπ‘₯=π‘˜π‘¦π‘§


True or False: If 𝑧 varies jointly with π‘₯ and 𝑦, and 𝑧=40 when 𝑦=2 and π‘₯=4, then the constant of variation equals 5.

  • ATrue
  • BFalse


Given that 45π‘₯βˆ’6𝑦3π‘₯βˆ’6𝑧=𝑦𝑧 and π‘₯βˆˆβ„οŠ°, which of the following relations is correct?

  • Aπ‘¦βˆπ‘§οŠ¨
  • Bπ‘¦βˆ1𝑧
  • Cπ‘¦βˆπ‘§
  • Dπ‘¦βˆ1π‘§οŠ¨


A variable 𝑧 varies jointly with variables π‘₯ and 𝑦. Given that 𝑧=12 when 𝑦=2 and π‘₯=3, write the equation that describes how 𝑧 depends on π‘₯ and 𝑦.

  • A𝑧=4π‘₯𝑦
  • B𝑧=3π‘₯𝑦
  • C𝑧=2π‘₯𝑦
  • D𝑧=12π‘₯𝑦
  • E𝑧=6π‘₯𝑦


True or False: Given that 𝑦 varies jointly with π‘₯ and 𝑧 such that 𝑦=2π‘₯𝑧, 𝑦=12 when π‘₯=2 and 𝑧=3.

  • ATrue
  • BFalse


When a body is moving onward, its kinetic energy 𝑇 varies jointly with mass π‘š and the square of velocity 𝑣. If 𝑇=9J when π‘š=2kg and 𝑣=3/ms, find 𝑣 when 𝑇=4J and π‘š=2kg.

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