Given that x is directly proportional to y and inversely proportional to Z, x = 15 when y = 10 and Z = 4, find the equation connecting x, y and z

Analyze the question: x varies directly as y and inversely as z. This is joint variation, so we combine both relationships and find the constant.

Concept:

• Direct proportion: x ∝ y
• Inverse proportion: x ∝ 1/z
• Combined: x ∝ y/z, so x = ky/z, where k is the constant of proportionality.

Solution:

Substitute x = 15, y = 10, z = 4 into x = ky/z:

15 = k(10)/4

15 = 10k/4

Multiply both sides by 4:

60 = 10k

k = 6

So the equation is:

x = 6y/z

Check: Put y = 10 and z = 4 back: x = 6(10)/4 = 60/4 = 15 ✓

Correct option: A

Tip: In variation questions, always follow the same steps: (1) write the proportionality, (2) introduce k, (3) substitute the given values to find k, (4) write the final equation with k. You can also check quickly by substituting the given values into each option, as we did above.