GCSE Ratios, fractions and linear functions Practice Questions

Read the ratio

For every 4 parts of x there are 3 parts of y, so every point on the line has the form (x, y) = (4t, 3t).

Divide y by x to get the multiplier

The multiplier is y divided by x — the y-part of the ratio over the x-part, not the other way round.

State the answer

y is three quarters of x. Note the 3 (the y-part) is on top.

Write the sentence as an equation

"y is 1/4 of x" means multiply x by 1/4 to get y.

Compare with y = kx

The number multiplying x is the gradient.

State the answer

A gradient of 1/4: 4 across for every 1 up.

Read the ratio

For every 7 parts of x there are 3 parts of y, so every point on the line has the form (x, y) = (7t, 3t).

Divide y by x to get the multiplier

The multiplier is y divided by x — the y-part of the ratio over the x-part, not the other way round.

Find y when $x = 21$

A seventh of $21$ is $3$, and $3 \times 3 = 9$.

Add the two values

The scale factor is 3, so the total is 3 x $(7 + 3) = 30$.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 7$ must give $y = 3$. It does, so the multiplier 3/7 is the right way round.

State the answer

$x + y = 30$.

Read the ratio

For every 5 parts of x there are 2 parts of y, so every point on the line has the form (x, y) = (5t, 2t).

Use a scale factor t

Points on the line are the ratio pair scaled by t.

Substitute into the condition

Replace x and y in $y = x - 9$.

Rearrange

Take 2t from both sides.

Solve for t

Divide both sides by 3.

Find the coordinates

$x = 5$ x 3 and $y = 2$ x 3.

Check both conditions

Both hold, so the point is correct.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 5$ must give $y = 2$. It does, so the multiplier 2/5 is the right way round.

Rule out the inverted ratio

The classic slip is to use 5/2 as the multiplier. Substituting $x = 5$ would then give $y = 25/2$, not 2, so it is wrong.

State the answer

The point is (15, 6).

Write the sentence as an equation

"y is 7/4 of x" means multiply x by 7/4 to get y.

Identify the gradient

The graph is a straight line through the origin with gradient $7/4 = 1.75$.

Ratio form

Taking $x = 4$ gives $y = 7$, so the ratio is 4 : 7.

Substitute $x = 12$

Use the equation for the first part.

Work out y

A quarter of $12$ is $3$, and $3 \times 7 = 21$, so the point is $(12, 21)$.

Check the ratio there

Dividing both by 3 gives back 4 : 7.

Now substitute $y = 63$

For the second part the y-value is known.

Rearrange

Multiply by the reciprocal of 7/4.

Work out x

$63 \times 4 = 252$, and $252$ divided by $7$ is $36$.

Check

A quarter of $36$ is $9$, and $9 \times 7 = 63$.

Check the ratio

Dividing both by 9 gives 4 : 7.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 4$ must give $y = 7$. It does, so the multiplier 7/4 is the right way round.

Rule out the inverted ratio

The classic slip is to use 4/7 as the multiplier. Substituting $x = 4$ would then give $y = 16/7$, not 7, so it is wrong.

Read the multiplier off the graph

From the origin, going 4 across and 7 up lands on the line, so the gradient is 7/4 — the same number as the multiplier.

State the answer

$x = 36$ when $y = 63$.