Variation · Form 5
Variation: Worked Examples (Medium)
This set moves into joint variation and combined direct-and-inverse variation, where the constant k is found from one full set of values and then used to answer a follow-up. It helps students who can do single variation but stumble when two quantities act at once.
Worked example 1
z varies directly as x and inversely as y. When x = 8 and y = 2, z = 20.
Find z when x = 6 and y = 5.
- Combined variation, so write z = kx/y.
- Substitute x = 8, y = 2, z = 20: 20 = k × 8 / 2.
- Simplify the right side: 20 = 4k.
- Solve for k: k = 20 ÷ 4 = 5.
- Equation is z = 5x/y.
- Substitute x = 6, y = 5: z = 5 × 6 / 5 = 30 ÷ 5 = 6.
Worked example 2
w varies directly as the square of r. When r = 3, w = 45.
Find the value of r when w = 125.
- Direct variation with a square, so write w = kr².
- Substitute r = 3, w = 45: 45 = k × 3² = 9k.
- Solve for k: k = 45 ÷ 9 = 5.
- Equation is w = 5r².
- Substitute w = 125: 125 = 5r², so r² = 125 ÷ 5 = 25.
- Take the positive square root: r = √25 = 5.
Worked example 3
F varies jointly as m and a. When m = 4 kg and a = 3 m/s², F = 24 N.
Find F when m = 7 kg and a = 5 m/s².
- Joint variation, so write F = kma.
- Substitute m = 4, a = 3, F = 24: 24 = k × 4 × 3.
- Simplify: 24 = 12k.
- Solve for k: k = 24 ÷ 12 = 2.
- Equation is F = 2ma.
- Substitute m = 7, a = 5: F = 2 × 7 × 5 = 70.
Worked example 4
z varies jointly as x and y. When x = 3 and y = 4, z = 60.
Find z when x = 5 and y = 6.
- Joint variation: z = kxy.
- Find k: 60 = k × 3 × 4 = 12k, so k = 5.
- Equation: z = 5xy.
- When x = 5 and y = 6: z = 5 × 5 × 6 = 150.
Worked example 5
p varies inversely as the square of q. When q = 2, p = 9.
Find p when q = 3.
- Inverse-square variation: p = k/q².
- Find k: 9 = k/2² = k/4, so k = 36.
- Equation: p = 36/q².
- When q = 3: p = 36 ÷ 3² = 36 ÷ 9 = 4.
Worked example 6
V varies directly as h and the square of r. When h = 4 and r = 3, V = 72.
Find V when h = 5 and r = 2.
- Combined variation: V = k h r².
- Find k: 72 = k × 4 × 3² = k × 4 × 9 = 36k, so k = 2.
- Equation: V = 2 h r².
- When h = 5 and r = 2: V = 2 × 5 × 2² = 2 × 5 × 4 = 40.
Worked example 7
y varies directly as the square of x and inversely as z. When x = 4 and z = 8, y = 10.
Find the value of y when x = 6 and z = 12.
- Write the combined equation: y = kx²/z.
- Substitute x = 4, z = 8, y = 10: 10 = k(4)²/8 = 16k/8 = 2k.
- Solve for k: k = 10 ÷ 2 = 5.
- Equation is y = 5x²/z.
- Substitute x = 6, z = 12: y = 5(6)²/12 = 5 × 36/12 = 180/12 = 15.
Worked example 8
P varies jointly as x and the cube of y. When x = 3 and y = 2, P = 48.
Find the value of P when x = 5 and y = 3.
- Write the joint equation: P = kxy³.
- Substitute x = 3, y = 2, P = 48: 48 = k(3)(2)³ = k × 3 × 8 = 24k.
- Solve for k: k = 48 ÷ 24 = 2.
- Equation is P = 2xy³.
- Substitute x = 5, y = 3: P = 2(5)(3)³ = 2 × 5 × 27 = 270.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What's tested at the medium level for variation?
Medium questions move to joint variation, where a quantity depends on two others at once, for example y varies directly as x and inversely as z (y = kx/z). You need to find k from one set of values, then use the same k to solve a related problem with new values.
What does the examiner reward in a joint variation question?
Marks are given for correctly setting up the joint variation equation first, then substituting the given values to find k as a clear separate step, before using that k to answer the actual question. Skipping straight to an answer without showing the value of k found usually loses method marks.
What's a common pitfall with joint variation questions?
Students often place a variable on the wrong side of the equation, for example writing y = kxz when z should be in the denominator, because they misread "inversely" in a sentence describing a joint relationship. Rewrite the relationship in words first before converting it into an equation.