Hard GCSE Ratios, fractions and linear functions Questions

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 $y = 63$.

Gradient of line A

From x : $y = 3$ : 8, the multiplier is the y-part over the x-part.

Check line A

Substituting $x = 3$ gives $y = 8$, matching the ratio.

Gradient of line B

From x : $y = 5$ : 12, the multiplier is 12/5.

Check line B

Substituting $x = 5$ gives $y = 12$.

Compare as decimals

Line A has the greater gradient, so line A is steeper.

Choose a common denominator

Fifteen is the lowest common denominator.

Rewrite line A

Multiply top and bottom by 5.

Rewrite line B

Multiply top and bottom by 3.

Subtract

Take the smaller gradient from the larger.

Check the fraction is simplest

$4 = 2 \times 2$ and $15 = 3 \times 5$, so nothing cancels.

Sense check with decimals

4/15 is about 0.267, which matches the decimal difference.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 3$ must give $y = 8$. It does, so the multiplier 8/3 is the right way round.

Rule out the inverted ratio

The classic slip is to use 3/8 as the multiplier. Substituting $x = 3$ would then give $y = 9/8$, not 8, so it is wrong.

Read the multiplier off the graph

From the origin, going 3 across and 8 up lands on the line, so the gradient is 8/3 — the same number as the multiplier.

State the answer

The difference in gradients is 4/15.

Read the ratio

For every 5 parts of x there are 3 parts of y, so every point on the line has the form (x, y) = (5t, 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.

Write the equation of the line

A ratio through the origin becomes y = kx with $k = 3/5$.

Substitute $y = 21$

Put the known y-value into the equation of the line.

Rearrange

Multiply by the reciprocal of 3/5.

Work out x

$21 \times 5 = 105$, and $105$ divided by $3$ is $35$.

State the point

The point on the line with $y = 21$ is (35, 21).

Check the ratio

Dividing both by 7 gives back 5 : 3.

Add the coordinates

The scale factor is 7, so the total is 7 x $(5 + 3) = 56$.

Check with the parts

Five parts plus three parts is eight parts, each worth 7.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 5$ must give $y = 3$. It does, so the multiplier 3/5 is the right way round.

Rule out the inverted ratio

The classic slip is to use 5/3 as the multiplier. Substituting $x = 5$ would then give $y = 25/3$, not 3, so it is wrong.

Read the multiplier off the graph

From the origin, going 5 across and 3 up lands on the line, so the gradient is 3/5 — the same number as the multiplier.

Unitary check

One unit of x carries 3/5 of y; every other point is just this scaled up.

State the answer

$x + y = 56$.

Use the point given

The line passes through the origin and through (9, 6), so those two coordinates are in the ratio 9 : 6.

Work out the gradient properly

Gradient is rise over run: y-coordinate over x-coordinate.

Test Beth's gradient

If the gradient were 3/2, $x = 9$ would give $y = 13.5$, not 6.

Test the correct gradient

The multiplier 2/3 reproduces the point exactly.

Find the correct ratio

Cancelling by 3 gives x : $y = 3$ : 2.

Compare with Beth

Beth has both the gradient and the ratio the wrong way round.

Note the trap

The gradient is the y-part over the x-part, so the ratio reverses the fraction.

Draw the line through the origin

Starting at the origin, a run of 9 and a rise of 6 reaches the line, so the gradient is 2/3.

Restate the three forms

Ratio, equation and gradient must all agree with the point (9, 6).

Check a second point

Doubling (9, 6) still fits, confirming the line.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 9$ must give $y = 6$. It does, so the multiplier 2/3 is the right way round.

Rule out the inverted ratio

The classic slip is to use 3/2 as the multiplier. Substituting $x = 9$ would then give $y = 27/2$, not 6, so it is wrong.

Read the multiplier off the graph

From the origin, going 9 across and 6 up lands on the line, so the gradient is 2/3 — the same number as the multiplier.

Unitary check

One unit of x carries 2/3 of y; every other point is just this scaled up.

State the answer

Both of Beth's statements are inverted.

Read the ratio

Every 2 m of fabric costs £9.

Write the multiplier

Cost divided by length is $9/2 = 4.5$ pounds per metre.

Write the equation

The cost graph is a straight line through the origin of gradient 4.5.

Interpret the gradient

The gradient is the price of one metre: £4.50.

Substitute $L = 7.5$

Multiply the length by the unit price.

Work out the product

$4.5 \times 7 = 31.5$ and $4.5 \times 0.5 = 2.25$; adding gives $33.75$.

Check by reversing

Dividing the cost by 4.5 gives back 7.5 m.

Check the ratio

Dividing both by 3.75 gives 2 : 9.

Sense check

7.5 m is a bit under 4 lots of 2 m, and £33.75 is a bit under 4 lots of £9 = £36.

Test the direction of the ratio

Substitute the ratio pair itself: $x = 2$ must give $y = 9$. It does, so the multiplier 9/2 is the right way round.

Rule out the inverted ratio

The classic slip is to use 2/9 as the multiplier. Substituting $x = 2$ would then give $y = 4/9$, not 9, so it is wrong.

Read the multiplier off the graph

From the origin, going 2 across and 9 up lands on the line, so the gradient is 9/2 — the same number as the multiplier.

Unitary check

One unit of x carries 9/2 of y; every other point is just this scaled up.

Check a second point on the line

Doubling the ratio pair gives another point on the same line, and it satisfies the equation, confirming the line passes through the origin.

State the answer

The cost is £33.75.