Direct and inverse proportion – Year 9 maths lesson to download as PDF

Worked example 1: finding \(k\)

\(y\) is directly proportional to \(x\), and \(y = 20\) when \(x = 4\). Find \(y\) when \(x = 9\).

\(y = kx\), so \(20 = k \times 4\), giving \(k = 5\). The equation is \(y = 5x\). When \(x = 9\), \(y = 5 \times 9 = 45\).

Worked example 2: the unitary method

5 pens cost £3.50. Find the cost of 8 pens.

The cost of one pen is \(3.50 \div 5 = £0.70\), so 8 pens cost \(8 \times 0.70 = £5.60\). This is the unitary method, which is the same idea as finding \(k\).

Worked example 3: inverse proportion

\(y\) is inversely proportional to \(x\), and \(y = 6\) when \(x = 4\). Find \(y\) when \(x = 8\).

\(y = \dfrac{k}{x}\), so \(6 = \dfrac{k}{4}\) and \(k = 24\). The equation is \(y = \dfrac{24}{x}\). When \(x = 8\), \(y = \dfrac{24}{8} = 3\). Doubling \(x\) from 4 to 8 halved \(y\) from 6 to 3, as expected.

Worked example 4: workers and days

It takes 6 workers 10 days to build a wall. How long would 4 workers take?

Total work is \(6 \times 10 = 60\) worker-days. With 4 workers: \(60 \div 4 = 15\) days. More workers means fewer days, so this is inverse proportion (assuming everyone works at the same rate).

Worked example 5: reading a table

\(x\): 1, 2, 4, 8 and \(y\): 16, 8, 4, 2. Which type of proportion is this?

The products \(xy\) are \(16, 16, 16, 16\), so this is inverse proportion with \(k = 16\). The equation is \(y = \dfrac{16}{x}\).

Worked example 6: an equation from one pair

The cost \(C\) of a taxi ride is directly proportional to the distance \(d\) in miles. A 7-mile ride costs £14. Find a formula and the cost of an 11-mile ride.

\(C=kd\). Substituting, \(14=7k\), so \(k=2\) and \(C=2d\). For 11 miles: \(C=2\times11=\) £22.

Worked example 7: testing a table

Is \(y\) in direct or inverse proportion to \(x\) in this table?

Table: \(x=1,2,4,8\) and \(y=24,12,6,3\). Check the products: \(1\times24=24\), \(2\times12=24\), \(4\times6=24\), \(8\times3=24\). The product is constant, so \(y\) is inversely proportional to \(x\), with \(y=\dfrac{24}{x}\).

Worked example 8: a mixed problem

8 machines make 600 toys in 5 hours. How long would 10 machines take to make 900 toys?

First find the rate of one machine: \(600\div8\div5=15\) toys per hour. Ten machines make \(10\times15=150\) toys per hour. So 900 toys take \(900\div150=\) 6 hours.

Worked example 9: reading a conversion

A recipe for 4 people needs 300 g of rice. How much rice is needed for 10 people, and how many people can be fed with 1.5 kg?

Rice per person \(=300\div4=75\) g. For 10 people: \(10\times75=750\) g. For 1.5 kg \(=1500\) g: \(1500\div75=20\) people. Both answers use the same constant, 75 g per person, which is the constant of proportionality.

Worked example 10: best buys

A 400 g jar of honey costs £3.20 and a 650 g jar costs £4.94. Which jar is better value?

The price of honey is directly proportional to its mass within each jar, so we can compare the cost of the same amount. Find the cost of 1 g in each jar. For the small jar, \(3.20\div400=0.008\) pounds, which is 0.8p per gram. For the large jar, \(4.94\div650=0.0076\) pounds, which is 0.76p per gram. The large jar is cheaper per gram, so it is the better buy. You could also compare the mass you get for £1: \(400\div3.20=125\) g and \(650\div4.94\approx131.6\) g, which gives the same conclusion.

Worked example 11: sharing the work

It takes 4 identical taps 9 minutes to fill a paddling pool. How long would 6 taps take?

More taps fill the pool faster, so the time goes down when the number of taps goes up: this is inverse proportion. The total amount of work is always the same. Four taps for 9 minutes is \(4\times9=36\) tap-minutes. Share this between 6 taps: \(36\div6=6\) minutes. A sensible check is that the answer must be less than 9 minutes, which it is.