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Direct and inverse proportion GCSE lesson: y = kx and y = k/x with graphs

Name: ____________________ Date: ____________________

Show your working. Check every root by substitution and label your sketches.

Proportion: practice

Show your working on paper: find k first. The stretch is optional.

Diagnostic question: choose and explain

  1. y is inversely proportional to x. When x = 2, y = 10. What is y when x = 4?
    1. 5
    2. 20
    3. 12
    4. 8

Practice questions

  1. y is directly proportional to x. y = 15 when x = 3. Find y when x = 7.
  2. y = kx, and y = 18 when x = 6. What is k?
  3. y is inversely proportional to x. y = 4 when x = 6. Find y when x = 8.
  4. Which table shows inverse proportion?
    1. x: 1, 2, 3 and y: 5, 10, 15
    2. x: 1, 2, 3 and y: 12, 6, 4
    3. x: 1, 2, 3 and y: 10, 9, 8
  5. Which graph shows y directly proportional to x?
    1. A straight line through (0, 0)
    2. A straight line crossing the y-axis at 3
    3. A curve that gets closer and closer to both axes
  6. y = 12/x. x is multiplied by 3. What happens to y?
    1. It is divided by 3
    2. It is multiplied by 3
    3. It goes down by 3
  7. A journey takes 3 hours at 40 mph. How many hours does the same journey take at 60 mph?
  8. Stretch: y is proportional to x². y = 12 when x = 2. Find y when x = 3.

Support: For question 3: work out k = x × y from the pair you are given. Then y = k/x for the new x. Check: x × y is the same as before.

Proportion: exit

One or two sentences are enough.

  1. Give a real example of two things in inverse proportion, and say why doubling one halves the other.

Exit check

Two sketches and one example show who tells the two kinds apart.

Sketch y = 3x and y = 12/x for x from 0 to 6 on one pair of axes, and mark where they meet.

Proportion: find k and use it

Say first whether each is direct or inverse. Find k, write the formula, then answer. Check each answer by working out k again from it.

1. y is directly proportional to x. y = 20 when x = 4. Find y when x = 9.

2. y is directly proportional to x. y = 14 when x = 2. Find x when y = 35.

3. y is inversely proportional to x. y = 9 when x = 4. Find y when x = 6.

4. 5 workers build a wall in 12 days. How long would 6 workers take, working at the same rate?

5. Sketch y = 2x and y = 18/x for x from 0 to 6 on one pair of axes. Where do they meet?

6. x: 2, 4, 8 and y: 16, 8, 4. Direct, inverse or neither? Write the formula, and find y when x = 16.

SKRM Learn · Original teaching resource · Free to use with your students.