GCSE Ratio notation and simplifying Practice Questions

Look for a common factor

Common factors of $4$ and $6$: $1$, $2$. The highest is $2$.

Divide both parts by the highest common factor

Dividing both parts of a ratio by the same number does not change the ratio.

Write the simplified ratio

$2$ and $3$ have no common factor left, so this is the simplest form.

Look for a factor common to all three parts

$2$ divides into $2$, $4$ and $6$ exactly, and it is the highest number that does.

Divide every part by the highest common factor

All three parts must be divided by the same number, or the ratio changes.

Write the simplified ratio

The three parts now share no common factor, so this is simplest form.

Find the flaw in the statement

The two amounts are in different units, so the numbers $20$ and $1$ cannot be compared directly. The student has ignored the units.

State the conversion fact

Centimetres are the smaller unit, so change the metre into centimetres.

Convert both amounts to the same unit

The second amount is $100$ cm, and now both parts share the unit centimetres.

Drop the units and write the ratio

Once both parts are in the same unit, the ratio is just a comparison of the two numbers.

Simplify the ratio

The highest common factor of $20$ and $100$ is $20$, and dividing gives $1$ : $5$.

Sense check the student answer

The student answer of 20 : 1 would mean the first amount is twenty times the second, but 20 cm is much SMALLER than 1 m. The correct ratio, 1 : 5, correctly shows the second amount is the bigger one.

Read the ratio

The numbers in a ratio are parts. They are not percentages and not totals.

Add the parts to find the whole

The whole group is made of $10$ equal parts.

Write the wanted group as a fraction of the whole

The prize b are 3 of the 10 parts.

Simplify the fraction

The fraction is already in its simplest form.

Convert the fraction to a percentage

A percentage is a number out of $100$, so multiply the fraction of the whole by $100$.

Find the scale factor that makes the total $100$

Scaling the ratio so that the parts total $100$ turns each part directly into a percentage.

Scale the whole ratio to a total of $100$

Every part is now written out of $100$, so each number is that group as a percentage.

Read off the percentage

The prize b make up 30 percent of the total.

Check the percentages add to $100$

Every member of the group is counted exactly once, so the percentages must total $100$.

Avoid the classic mistake

Comparing the wanted part with the OTHER part instead of with the total is the usual error here. A percentage of the whole needs the TOTAL on the bottom.

Write down the two ratios

Two separate ratios share the quantity Year $11$, which is the link between them.

Spot the linking quantity

Year $11$ appears in both ratios but as a different number of parts, so the two ratios cannot simply be written side by side yet.

Find the lowest common multiple of the two linking numbers

$10$ is the smallest number that both $5$ and $10$ divide into, so make Year $11$ worth $10$ parts in BOTH ratios.

Scale the first ratio

Multiply both parts of the first ratio by $2$ so that Year $11$ becomes $10$.

Scale the second ratio

Year $11$ is already worth $10$ parts here, so the second ratio does not need scaling.

Check the linking quantity now matches

Year $11$ is worth $10$ parts in both ratios, so the two ratios can now be joined.

Combine into a three-part ratio

Take Year $10$ from the first ratio, the shared Year $11$, and Year $12$ from the second.

Check the three-part ratio is in simplest form

The three parts share no common factor, so no cancelling is needed.

Check the first ratio is still there

Cancelling the first two parts of the answer returns the original Year $10$ to Year $11$ ratio, as it must.

Check the second ratio is still there

Cancelling the last two parts returns the original Year $11$ to Year $12$ ratio, so the combined ratio agrees with both starting ratios.

Warn against the classic mistake

Writing the given numbers side by side ignores that Year $11$ counts differently in each ratio. The linking part must be scaled to match first.

Find the difference in parts

In the combined ratio there are $2$ more parts of Year $11$ than of Year $10$.

Match the difference in parts to the difference given

The question says there are $40$ more Year $11$ than Year $10$, and that gap is worth $2$ parts.

Work out the value of one part

Dividing the difference by the number of parts in the difference gives the size of a single part.

Multiply by the total number of parts

The whole group is $21$ parts and each part is worth $20$.