Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 8 of 8

What stays the same when two quantities move together?

In every proportional relationship one combination of the two quantities is constant. Find that constant from a single pair of values and the whole relationship follows, including its graph.

Number and AlgebraGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 8: 5 books cost 40. How much do 8 books cost?
    Answer64. Each book costs 8.
  2. From lesson 6: make x the subject of y = 6x.
    Answerx = y ÷ 6.
  3. From Year 9: evaluate 4x² when x = 3.
    Answer36.
Words used in this lesson
direct proportion
two quantities whose ratio is always the same. For example: y = 9x.
inverse proportion
two quantities whose product is always the same. For example: y = 20 ÷ x.
constant of proportionality
the fixed number k that links the two quantities. For example: the 9 in y = 9x.
origin
the point (0, 0) where the axes cross. For example: (0, 0).
proportional to the square of
equal to a constant times the square. For example: area of a circle and radius.

Do twice the cooks mean half the time?

A kitchen in Istanbul prepares a banquet. With 6 cooks the work takes 8 hours. The head chef says that if she hires 12 cooks the banquet will be ready much sooner. Her manager asks exactly how much sooner, and what has to be true for her answer to hold.

Predict first: how long would 12 cooks need, and what are you assuming about the cooks? Write both down. Then use the figure: the point moves along four kinds of relationship, and one combination of x and y stays constant.

y = kx
2
2
4y at that x
y ÷ x = 2what stays constant

y = kx with k = 2: at x = 2, y = 4 and y ÷ x = 2, the value of k.

Show me the steps
  1. y = kx, k = 2, x = 2: y = 4 and y ÷ x = 2. The graph is a straight line through the origin, and doubling x doubles y.
  2. y = k/x, k = 6, x = 2: y = 3 and y × x = 6. The curve falls, and doubling x halves y.
  3. y = kx², k = 2, x = 3: y = 18 and y ÷ x² = 2. Doubling x multiplies y by 4.
  4. In every case one combination of x and y equals k: find k from one pair, then use it for every other pair.

Direct and inverse proportion each have a constant k. In direct proportion y = kx, so y ÷ x is always k and the graph is a straight line through the origin. In inverse proportion y = k/x, so y × x is always k. A power or root changes the combination: y = kx² has y ÷ x² constant; y = k/x² has y × x² constant.

Method: find k from one pair of values, write the equation, then substitute.

Two common mistakes. Calling any rising relationship direct: y = x + 5 rises, but y ÷ x is not constant and its graph misses the origin. And using direct proportion for a fall: if y falls as x rises, test y × x; for y = 12 − x the product is not constant, so it is not inverse proportion either.

Worked examples

1. y is directly proportional to x. y = 21 when x = 3. Find y when x = 10.

  1. y = kx, so 21 = 3k and k = 7.
  2. y = 7 × 10 = 70.

2. The time t for a journey is inversely proportional to the speed v. At 60 km/h it takes 5 hours. How long at 100 km/h?

  1. t = k/v, so 5 = k/60 and k = 300.
  2. t = 300 ÷ 100 = 3 hours.

3. The mass m of a solid metal cube is directly proportional to the cube of its side s. A cube of side 2 cm has mass 16 g. Find the mass of a cube of side 5 cm.

  1. m = ks3, so 16 = k × 8 and k = 2.
  2. m = 2 × 53 = 2 × 125 = 250 g.

Your go. The first step is given. The brightness B of a lamp is inversely proportional to the square of the distance d. At 2 m the brightness is 90 units. Find it at 3 m. The first step: B = k/d2, so 90 = k/4.

  1. 90 = k/4.
  2. Find k, then substitute d = 3.
Check the answer

k = 90 × 4 = 360. At 3 m, B = 360 ÷ 9 = 40 units.

With a GDC (optional)

A GDC can plot a proportional relationship. In the graph menu enter the equation, for example y = 5/x, and look at the curve. Use the trace feature to read pairs of values and multiply each pair: for inverse proportion the products should all be the same number.

The graph is a check, not a method. To find a value you still find k from one pair and substitute. Everything here can be done by hand.

Check before you go on

1. y is inversely proportional to x. When x = 4, y = 6. What is y when x = 8?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. y is directly proportional to x. When x = 7, y = 35. What is y when x = 12?

3. y is inversely proportional to x. When x = 3, y = 12. What is y when x = 9?

Level 2 · Building

4. y is directly proportional to x². When x = 2, y = 32. What is y when x = 6?

5. The time to fill a tank is inversely proportional to the number of pumps used. Three pumps take 40 minutes. How many minutes do 8 pumps take?

Level 3 · Stretch

6. y is inversely proportional to x². When x = 3, y = 8. What is y when x = 6?

7. y is directly proportional to the square root of x. When x = 4, y = 6. What is x when y = 15?

Something to think about

Compare y = 3x with y = 3 × 2x for x = 1, 2, 3, 4. Which of the two is directly proportional to x? What does adding 1 to x do to y in each case?

One way to think about it

y = 3x gives 3, 6, 9, 12: y ÷ x is always 3, so it is directly proportional, and adding 1 to x adds 3 to y. y = 3 × 2x gives 6, 12, 24, 48: y ÷ x is 6, 6, 8, 12, which is not constant, so it is not proportional, and adding 1 to x multiplies y by 2. A constant ratio between neighbouring terms is geometric growth, not proportion.

Exit ticket

8. Which table shows y inversely proportional to x?