GCSE Direct and inverse proportion Practice Questions

Identify the type of proportion

Directly proportional means $y = kx$: multiply $x$ by a number and $y$ is multiplied by the same number.

Write the general formula

Here $k$ is the constant of proportionality. It is the same for every pair of values in the relationship.

Divide y by x to find k

For direct proportion $k = \frac{y}{x}$, so $k = 20 \div 4 = 5$.

Decide which kind of proportion this is

More taps fill the tank in less time, so time and number of taps are inversely proportional.

Work out the total tap-minutes

Filling the tank takes $120$ tap-minutes, so $k = 120$.

Divide by the new number of taps

So $10$ taps fill the tank in $12$ minutes.

Identify the type of proportion

Directly proportional means $y = kx$: multiply $x$ by a number and $y$ is multiplied by the same number.

Write the general formula

Here $k$ is the constant of proportionality. It is the same for every pair of values in the relationship.

Substitute the pair of values you are given

Put $x = 15$ and $y = 6$ into $y = kx$.

Divide y by x

For direct proportion $k = \frac{y}{x}$.

Simplify the fraction

Divide the top and the bottom by $3$.

State the constant

The constant of proportionality is $\frac{2}{5}$, so $y = \frac{2}{5}x$.

Identify the type of proportion

Inversely proportional means $y = \frac{k}{x}$: multiply $x$ by a number and $y$ is divided by that number.

Write the general formula

Here $k$ is the constant of proportionality. For inverse proportion the product $x \times y$ is always the same.

Substitute the pair of values you are given

Put $x = 9$ and $y = 4$ into $y = \frac{k}{x}$.

Solve for the constant of proportionality

Multiplying both sides by $9$ gives $k = 36$.

Write the formula for this relationship

Every pair of values in this relationship has $x \times y = 36$.

Check the formula against the given pair

It gives back $y = 4$ when $x = 9$, so the value of $k$ is right.

Substitute the new value of x

Now use $x = 3$ in $y = \frac{36}{x}$.

Work out the value of y

So $y = 12$ when $x = 3$.

Subtract the old value of y from the new one

The increase is the new value of $y$ minus the old value.

Check with the multiplier

$x$ has been multiplied by $\frac{1}{3}$, so in inverse proportion $y$ is divided by $\frac{1}{3}$. This agrees with the answer.

Decide which kind of proportion this is

Fewer workers take more hours, so hours and workers are inversely proportional.

Write the inverse proportion formula

Here $h$ is the number of hours and $w$ is the number of workers.

Substitute the given pair

Put $w = 5$ and $h = 12$ into the formula.

Find the constant, the total worker-hours

The floor needs $60$ worker-hours of work.

Write the formula for this job

Workers multiplied by hours is always $60$.

Check the formula against the given pair

It gives back $12$ hours, so the constant is right.

Substitute four workers

Now use $w = 4$.

Work out the time taken

Four workers take $15$ hours.

Check the direction of the change

Fewer workers take longer, so the answer makes sense.

Work out the total hours worked by all the workers

Each of the $4$ workers works for $15$ hours, giving $60$ worker-hours in total.

Notice the worker-hours are unchanged

The total worker-hours is the constant of proportionality, so it is the same either way.

Write the formula for the wage bill

Each worker-hour costs £$9$, so the bill is directly proportional to the total worker-hours.

Work out the wage bill

The total wage bill is £$540$.

Compare with five workers

Five workers for $12$ hours would cost the same, because the worker-hours are the same. Using fewer workers does not save money here.

State the final answer

The answer is $540$.