Sec 2 · G3 (Express)

Direct and inverse proportion

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1. Direct proportion

Left: y = 2x is a straight line through the origin, through (2, 4) and (5, 10). Right: y = 12 over x is a curve through (2, 6) and (6, 2); x times y stays 12.

Words	Equation
$y$ is directly proportional to $x$	$y = kx$
$y$ is directly proportional to $x^{2}$	$y = kx^{2}$
$y$ is directly proportional to $x^{3}$	$y = kx^{3}$
$y$ is directly proportional to $\sqrt{x}$	$y = k\sqrt{x}$

y = 12 when x = 4: find k first

\displaystyle y = kx

e.g.

\displaystyle 12 = k(4)

\displaystyle k = 3

so $y = 3x$ ; when $x = 7$ , $y = 21$

y is proportional to x², and y = 18 when x = 3

\displaystyle y = kx^{2}

e.g.

\displaystyle 18 = k(3)^{2}

\displaystyle k = 2

when $x = 5$ : $y = 2(5)^{2}$ , which is $50$

📖 Worked example

$y$ is directly proportional to $x$ . When $x = 2$ , $y = 16$ . Find the value of $y$ when $x = 12$ .

Step 1 of 3

Write the equation connecting y and x, with a constant k

y = kx

📖 Worked example

$y$ is directly proportional to the cube of $x$ . Given that $y = 32$ when $x = 2$ , express $y$ in terms of $x$ .

Step 1 of 3

Write the equation connecting y and x, with a constant k

y = kx^3

📖 Worked example

The graph of $y$ against $x$ is a straight line through the points $(4, 1)$ and $(10, 7)$ . Is $y$ directly proportional to $x$ ?

Step 1 of 2

Work out y ÷ x at each point

(4, 1)

\dfrac{y}{x} = \dfrac{1}{4}

\dfrac{y}{x} = \dfrac{1}{4}

.At

(10, 7)

\dfrac{y}{x} = \dfrac{7}{10}

\dfrac{y}{x} = \dfrac{7}{10}

📖 Worked example

The extension of a spring is directly proportional to the mass hung on it. A mass of 200 g stretches the spring by 3 cm. Find the extension, in millimetres, when a mass of 1.5 kg is hung on it.

Step 1 of 3

Change 1.5 kg to g, the unit in the pair given

1.5

= 1.5 \times 1000

= 1500

📖 Worked example

$y$ is directly proportional to the square of $x$ . When $x$ increases from $4$ to $6$ , $y$ increases by $10$ . Find the value of $y$ when $x = 2$ .

Step 1 of 4

Write the equation with a constant k

y = kx^2

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2. Inverse proportion

Words	Equation
$y$ is inversely proportional to $x$	$y = \dfrac{k}{x}$
$y$ is inversely proportional to $x^{2}$	$y = \dfrac{k}{x^{2}}$

y = 6 when x = 4

\displaystyle y = \dfrac{k}{x}

e.g.

\displaystyle 6 = \dfrac{k}{4}

\displaystyle k = 24

when $x = 8$ : $y = \dfrac{24}{8}$ , which is $3$

Workers and days: 6 workers take 10 days

\displaystyle \text{workers} \times \text{days} = 60

e.g.

\displaystyle \dfrac{60}{4}

\displaystyle = 15

$4$ workers take $15$ days

📖 Worked example

10 workers take 25 days to paint a block of flats. Find the number of days 5 workers would take, working at the same rate.

Step 1 of 2

Find how long ONE worker would take

\text{1 worker} \to 10 \times 25

= 250

days

📖 Worked example

6 machines, running 5 hours a day, take 16 days to make an order of bottles. Find the number of days 12 machines, running 10 hours a day, would take to make the same order.

Step 1 of 2

The whole job in machine-hours: machines × hours a day × days

\text{Total} = 6 \times 5 \times 16

= 480

machine-hours

📖 Worked example

A farmer has enough feed to last 14 cows for 10 days. After 7 days, the farmer sells 7 of the cows. Assuming that every cow eats the same amount each day, find the number of days the remaining feed will last.

Step 1 of 3

The assumption behind the method

We assume that every cow eats the same amount each day.

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3. Map scales

Scale	Length	Area
$1 : 50\,000$	$1\text{ cm} : 0.5\text{ km}$	$1\text{ cm}^{2} : 0.25\text{ km}^{2}$
$1 : 20\,000$	$1\text{ cm} : 0.2\text{ km}$	$1\text{ cm}^{2} : 0.04\text{ km}^{2}$

1 : 50 000, a 7.4 cm road on the map

\displaystyle 7.4 \times 0.5

e.g.

\displaystyle 7.4 \times 0.5

\displaystyle = 3.7

$3.7$ km on the ground

📖 Worked example

A map is drawn to a scale of $1 : 50\,000$ . Two towns are $5$ cm apart on the map. Find the actual distance, in kilometres, between the two towns.

Step 1 of 2

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 50\,000

\mathstrut 1\text{ cm} : 50\,000\text{ cm}

1\text{ cm} : 0.5\text{ km}

📖 Worked example

A map is drawn to a scale of $1 : 20\,000$ . A park has an area of $15$ cm² on the map. Find the actual area of the park, in km².

Step 1 of 3

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 20\,000

\mathstrut 1\text{ cm} : 20\,000\text{ cm}

1\text{ cm} : 0.2\text{ km}

📖 Worked example

A map is drawn to a scale of $1 : 20\,000$ . On the map, the route from a school to a sports centre is $15$ cm long. A cyclist leaves the school at $0955$ and must reach the sports centre at $1025$ . Find the average speed, in km/h, at which the cyclist must travel.

Step 1 of 4

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1\text{ cm} : 20\,000\text{ cm}

1\text{ cm} : 0.2\text{ km}

Practise this →

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Sec 2 · G3 (Express)

Direct and inverse proportion

Practise direct and inverse proportion with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Direct proportion

Words	Equation
$y$ is directly proportional to $x$	$y = kx$
$y$ is directly proportional to $x^{2}$	$y = kx^{2}$
$y$ is directly proportional to $x^{3}$	$y = kx^{3}$
$y$ is directly proportional to $\sqrt{x}$	$y = k\sqrt{x}$

y = 12 when x = 4: find k first

\displaystyle y = kx

e.g.

\displaystyle 12 = k(4)

\displaystyle k = 3

so $y = 3x$ ; when $x = 7$ , $y = 21$

y is proportional to x², and y = 18 when x = 3

\displaystyle y = kx^{2}

e.g.

\displaystyle 18 = k(3)^{2}

\displaystyle k = 2

when $x = 5$ : $y = 2(5)^{2}$ , which is $50$

📖 Worked example

$y$ is directly proportional to $x$ . When $x = 2$ , $y = 16$ . Find the value of $y$ when $x = 12$ .

Step 1 of 3

Write the equation connecting y and x, with a constant k

y = kx

📖 Worked example

$y$ is directly proportional to the cube of $x$ . Given that $y = 32$ when $x = 2$ , express $y$ in terms of $x$ .

Step 1 of 3

Write the equation connecting y and x, with a constant k

y = kx^3

📖 Worked example

The graph of $y$ against $x$ is a straight line through the points $(4, 1)$ and $(10, 7)$ . Is $y$ directly proportional to $x$ ?

Step 1 of 2

Work out y ÷ x at each point

(4, 1)

\dfrac{y}{x} = \dfrac{1}{4}

\dfrac{y}{x} = \dfrac{1}{4}

.At

(10, 7)

\dfrac{y}{x} = \dfrac{7}{10}

\dfrac{y}{x} = \dfrac{7}{10}

📖 Worked example

The extension of a spring is directly proportional to the mass hung on it. A mass of 200 g stretches the spring by 3 cm. Find the extension, in millimetres, when a mass of 1.5 kg is hung on it.

Step 1 of 3

Change 1.5 kg to g, the unit in the pair given

1.5

= 1.5 \times 1000

= 1500

📖 Worked example

$y$ is directly proportional to the square of $x$ . When $x$ increases from $4$ to $6$ , $y$ increases by $10$ . Find the value of $y$ when $x = 2$ .

Step 1 of 4

Write the equation with a constant k

y = kx^2

Practise this →

2. Inverse proportion

Words	Equation
$y$ is inversely proportional to $x$	$y = \dfrac{k}{x}$
$y$ is inversely proportional to $x^{2}$	$y = \dfrac{k}{x^{2}}$

y = 6 when x = 4

\displaystyle y = \dfrac{k}{x}

e.g.

\displaystyle 6 = \dfrac{k}{4}

\displaystyle k = 24

when $x = 8$ : $y = \dfrac{24}{8}$ , which is $3$

Workers and days: 6 workers take 10 days

\displaystyle \text{workers} \times \text{days} = 60

e.g.

\displaystyle \dfrac{60}{4}

\displaystyle = 15

$4$ workers take $15$ days

📖 Worked example

10 workers take 25 days to paint a block of flats. Find the number of days 5 workers would take, working at the same rate.

Step 1 of 2

Find how long ONE worker would take

\text{1 worker} \to 10 \times 25

= 250

days

📖 Worked example

6 machines, running 5 hours a day, take 16 days to make an order of bottles. Find the number of days 12 machines, running 10 hours a day, would take to make the same order.

Step 1 of 2

The whole job in machine-hours: machines × hours a day × days

\text{Total} = 6 \times 5 \times 16

= 480

machine-hours

📖 Worked example

Step 1 of 3

The assumption behind the method

We assume that every cow eats the same amount each day.

Practise this →

3. Map scales

Scale	Length	Area
$1 : 50\,000$	$1\text{ cm} : 0.5\text{ km}$	$1\text{ cm}^{2} : 0.25\text{ km}^{2}$
$1 : 20\,000$	$1\text{ cm} : 0.2\text{ km}$	$1\text{ cm}^{2} : 0.04\text{ km}^{2}$

1 : 50 000, a 7.4 cm road on the map

\displaystyle 7.4 \times 0.5

e.g.

\displaystyle 7.4 \times 0.5

\displaystyle = 3.7

$3.7$ km on the ground

📖 Worked example

A map is drawn to a scale of $1 : 50\,000$ . Two towns are $5$ cm apart on the map. Find the actual distance, in kilometres, between the two towns.

Step 1 of 2

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 50\,000

\mathstrut 1\text{ cm} : 50\,000\text{ cm}

1\text{ cm} : 0.5\text{ km}

📖 Worked example

A map is drawn to a scale of $1 : 20\,000$ . A park has an area of $15$ cm² on the map. Find the actual area of the park, in km².

Step 1 of 3

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1 : 20\,000

\mathstrut 1\text{ cm} : 20\,000\text{ cm}

1\text{ cm} : 0.2\text{ km}

📖 Worked example

Step 1 of 4

Write the scale as a ratio with units (1 km = 100 000 cm)

\mathstrut 1\text{ cm} : 20\,000\text{ cm}

1\text{ cm} : 0.2\text{ km}

Practise this →

Try a question: Direct and inverse proportion

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise direct and inverse proportion with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free