MathematicsExam code: 0580
Proportion
Algebra and Sequences
When it is not simply x
- The syllabus allows proportion to a square, a square root, a cube or a cube root, and the method never changes
- Only the equation in step one differs, so read the wording carefully and translate it exactly
| y is proportional to… | Equation to write down |
|---|---|
| x | y = kx |
| the square of x | y = kx² |
| the square root of x | y = k√x |
| the cube of x | y = kx³ |
| the cube root of x | y = k∛x |
| y is inversely proportional to… | Equation to write down |
|---|---|
| x | y = k ÷ x |
| the square of x | y = k ÷ x² |
| the square root of x | y = k ÷ √x |
| the cube of x | y = k ÷ x³ |
| the cube root of x | y = k ÷ ∛x |
- Questions frequently use a whole expression rather than a single letter, such as being proportional to (x + 2)² or to √(t + 2)
- In that case the bracket travels through the working untouched: substitute the value of the letter into the bracket first, then work out the power or root
Worked example
Proportional to a square
y is proportional to the square of x, where x > 0. When x = 4, y = 48. Find x when y = 108.
Solution:
- The wording gives y = kx²
- Substitute the given pair: 48 = k × 4², which is 48 = 16k
- Divide by 16: k = 3
- The finished equation is y = 3x²
- Now substitute y = 108: 108 = 3x²
- Divide by 3: x² = 36
- Square root: x = 6 or x = −6, and the question says x > 0
- So x = 6
Exam tip
Two method marks before the answer
When a question asks for one new value, the second method mark is for using your own k with the new value, so finish that step even if your k looks wrong. For inverse proportion, one line of equal products earns both method marks at once. In the next example y × (x + 3)² never changes, so 2 × (1 + 3)² = y × (5 + 3)² is that line.
Worked example
Inversely proportional to a bracket
y is inversely proportional to (x + 3)². When x = 1, y = 2. Find y when x = 5.
Solution:
- The wording gives y = k ÷ (x + 3)²
- Substitute the given pair: 2 = k ÷ (1 + 3)², so 2 = k ÷ 16
- Multiply by 16: k = 32
- The finished equation is y = 32 ÷ (x + 3)²
- Substitute x = 5: y = 32 ÷ (5 + 3)² = 32 ÷ 64
- So y = 0.5