Sec 2 · G1 (N(T))

Direct and inverse proportion

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

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 = 12 when x = 4: find k first

\displaystyle y = kx

e.g.

\displaystyle 12 = k(4)

\displaystyle k = 3

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

📖 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

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2. Inverse proportion

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

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

Practise this →

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1. Direct proportion

Words	Equation
$y$ is directly proportional to $x$	$y = kx$

y = 12 when x = 4: find k first

\displaystyle y = kx

e.g.

\displaystyle 12 = k(4)

\displaystyle k = 3

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

📖 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

Practise this →

2. Inverse proportion

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

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

Practise this →