y is inversely proportional to x2. When x = 2, y = 9. Find y when x = 3.
- Write the formula: y = k/x2.
- Find k: 9 = k/22 = k/4, so k = 9 × 4 = 36.
- Use the formula: y = 36/32 = 36/9 = 4.
Answer: y = 4
Proportion
Maths · Algebra and graphs. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Two quantities are in direct proportion when one is a fixed multiple of the other. If y is directly proportional to x, then y = kx, where k is a number called the constant of proportionality. When x doubles, y doubles.
In inverse proportion one quantity goes up as the other goes down, and their product stays the same: y = k/x. When x doubles, y halves.
The same idea works with powers and roots of x, such as x2, x3 or √x. The method never changes: write the formula with k, use the given pair of values to find k, then use the formula to find the unknown.
Rules to remember
- y directly proportional to x: y = kx
- y directly proportional to x2: y = kx2 (and y = kx3, y = k√x in the same way)
- y inversely proportional to x: y = k/x
- y inversely proportional to x2: y = k/x2 (and y = k/√x in the same way)
- The symbol ∝ means "is proportional to"
- If y = kx2, doubling x makes y 4 times as large; if y = k/x2, doubling x makes y one quarter as large
Common mistake
Students often use y = kx when the question says proportional to the square, or forget to square x when finding k. Underline the exact words in the question and write the formula before putting in any numbers.
Worked example
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