Direct proportion: y = kx

The Maths Habit
Walkthrough to print

When does doubling
one amount double
another?

The cost of apples and how many you buy

Pay and the hours you work

Distance and time at a steady speed

∝
Is proportional to.
k
The constant of proportionality: the number you multiply by.

y ∝ x

y = kx

Direct proportion

  1. StatementWrite it with ∝.
  2. Replace ∝ with = kSwap the ∝ for = k.
  3. SubstitutePut in the pair of values you are given.
  4. Find kDivide to find k.
  5. FormulaWrite the formula with k in.
  6. Use itPut in the new value and work it out.
y ∝ x
y = kx
y value = k × x value
k = y valuex value
y = kx
y = k × new x
∝ means is proportional to
k as a number
finding x? put in y and divide by k
Walkthrough to print

Write it as an equation

y ∝ x y = kx

Find k

y = kx,   x = 5, y = 30 6

  1. y ∝ x. When x = 4, y = 28. Write a formula for y in terms of x.

    y = 7x

  2. y ∝ x. When x = 5, y = 12. Find k.

    2.4

  3. A ∝ r. When r = 6, A = 45. Write a formula for A in terms of r.

    A = 7.5r

  4. Which formula shows that y is directly proportional to x:  y = x + 4  or  y = 4x?

    y = 4x

Find x when y = 40

y = 8x 5

y is directly proportional to x. When x = 6, y = 42. Find y when x = 9.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute42 = k × 64Find kk = 426 = 75Formulay = 7x6Use it, x = 9y = 7 × 9 = 63

y is directly proportional to x. When x = 5, y = 20. Find y when x = 12.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute20 = k × 54Find kk = 205 = 45Formulay = 4x6Use it, x = 12y = 4 × 12 = 48

y is directly proportional to x. When x = 6, y = 42. Find y when x = 9.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute42 = k × 64Find kk = 426 = 75Formulay = 7x6Use it, x = 9y = 7 × 9 = 63

y is directly proportional to x. When x = 8, y = 24. Find y when x = 11.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute24 = k × 84Find kk = 248 = 35Formulay = 3x6Use it, x = 11y = 3 × 11 = 33

y is directly proportional to x. When x = 6, y = 42. Find y when x = 9.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute42 = k × 64Find kk = 426 = 75Formulay = 7x6Use it, x = 9y = 7 × 9 = 63

y is directly proportional to x. When x = 12, y = 60. Find y when x = 7.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute60 = k × 124Find kk = 6012 = 55Formulay = 5x6Use it, x = 7y = 5 × 7 = 35

y is directly proportional to x. When x = 6, y = 42. Find y when x = 9.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute42 = k × 64Find kk = 426 = 75Formulay = 7x6Use it, x = 9y = 7 × 9 = 63

y is directly proportional to x. When x = 7, y = 56. Find x when y = 96.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute56 = k × 74Find kk = 567 = 85Formulay = 8x6Use it, y = 9696 = 8x  so  x = 12

y is directly proportional to x. When x = 6, y = 42. Find y when x = 9.

1Statementy ∝ x2Replace ∝ with = ky = kx3Substitute42 = k × 64Find kk = 426 = 75Formulay = 7x6Use it, x = 9y = 7 × 9 = 63

d is directly proportional to t. When t = 5, d = 17. Find d when t = 8.

1Statementd ∝ t2Replace ∝ with = kd = kt3Substitute17 = k × 54Find kk = 175 = 3.45Formulad = 3.4t6Use it, t = 8d = 3.4 × 8 = 27.2

Direct proportion: y = kx worksheet, page 1

Answer sheet

  1. The stretch of a spring is directly proportional to the mass on it. 200 g stretches it 5 cm. What mass stretches it 12 cm?

    480 g

  2. y ∝ x and y = 18 when x = 4. Find the value of x that makes y equal to x + 21.

    6

  1. y is directly proportional to x. When x = 6, y = 21. Write a formula for y in terms of x.

    y = 3.5x

  2. y is directly proportional to x. When x = 5, y = 35. Find y when x = 8.

    y = 56

  3. P is directly proportional to Q. When Q = 4, P = 30. Find Q when P = 75.

    Q = 10
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