Hard GCSE Ratio notation and simplifying Questions

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 $20$.

Write down the two ratios

Two separate ratios share the quantity q, which is the link between them.

Spot the linking quantity

q 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

15 is the smallest number that both 3 and 5 divide into, so make q worth 15 parts in BOTH ratios.

Scale the first ratio

Multiply both parts of the first ratio by 5 so that q becomes 15.

Scale the second ratio

Multiply both parts of the second ratio by 3 so that q becomes 15 here too.

Check the linking quantity now matches

q is worth 15 parts in both ratios, so the two ratios can now be joined.

Combine into a three-part ratio

Take p from the first ratio, the shared q, and r 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 p to q ratio, as it must.

Check the second ratio is still there

Cancelling the last two parts returns the original q to r ratio, so the combined ratio agrees with both starting ratios.

Warn against the classic mistake

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

Pick out the two parts that are wanted

The three-part ratio already contains the p to r ratio: just read off those two parts.

Simplify that pair

Both parts divide by $2$.

Divide both parts by the first part

To reach the form 1 : n the first part must be 1, so divide BOTH parts by 5.

Check by scaling back up

Scaling $1$ : $1.2$ back up returns $5$ : $6$, so the answer is consistent.

Read the ratio of fractions

Three fractions are being compared. A ratio is only in an acceptable form when its parts are whole numbers with no common factor.

Write down the denominators

The denominators are what stand in the way of whole numbers.

Find the lowest common multiple of the denominators

$12$ is the smallest number that all the denominators divide into exactly, so multiplying by $12$ clears every fraction at once.

Multiply part $1$ by the lowest common multiple

$12$ divided by $4$ is $3$, and $3$ times $3$ is $9$.

Multiply part $2$ by the lowest common multiple

$12$ divided by $6$ is $2$, and $2$ times $5$ is $10$.

Multiply part $3$ by the lowest common multiple

$12$ divided by $2$ is $6$, and $6$ times $1$ is $6$.

Write the ratio with whole numbers

Every part was multiplied by the same number, so the new ratio is equivalent to the old one.

Look for a common factor

The three parts share no common factor bigger than $1$.

Write the ratio in its simplest form

This is the simplified whole-number ratio.

Check the answer is in its simplest form

The parts share no common factor bigger than $1$, so the ratio cannot be cancelled any further.

Check the first two parts against the original fractions

The first fraction divided by the second gives the same value as the first part of the answer divided by the second, so those two parts are in the right proportion.

Check the last two parts as well

The same check works on the second and third parts, so the whole three-part ratio is correct.

Find the total number of parts

If these parts made up a whole, the whole would be $25$ parts.

Express the middle part as a fraction of the whole

The middle part is $10$ of the $25$ parts, which is $2/5$ of the total. A part is always taken out of the SUM of the parts.

Avoid the common mistake

Comparing only the numerators and ignoring the denominators gives a completely different, wrong ratio. The fractions must be cleared using the LCM.

Read the percentages given

Two of the three percentages are given directly in the question.

Find the missing percentage

Everyone else is in the third group, so subtract the two known percentages from $100$. That gives $40$ percent.

Check the three percentages total $100$

Each person is counted exactly once, so the three percentages must add to $100$.

Turn the percentages into parts

A percentage is already a number out of $100$, so the three percentages can be written straight down as a ratio.

Explain why that works

Each percentage is a fraction with denominator $100$, and multiplying all three parts by $100$ clears that denominator, leaving the percentages themselves.

Look for a common factor

$5$ divides into all three parts.

Divide every part by the highest common factor

Dividing all three parts by the same number gives an equivalent ratio.

Check the answer is in its simplest form

The parts share no common factor bigger than $1$, so the ratio cannot be cancelled any further.

Check the parts still describe the whole

The whole is now $20$ parts instead of $100$, but the proportions are unchanged.

Check the first group as a percentage

Turning the first part back into a percentage returns $25$ percent, exactly what the question gave.

Check the second group as a percentage

This returns $35$ percent, which also matches.

Check the third group as a percentage

This returns $40$ percent, the group that was left over.

Test the ratio with real numbers

With $200$ people the three groups would have $50$, $70$, $80$ members, and cancelling those numbers gives the same ratio.

Avoid the common mistake

Forgetting the leftover group and writing only a two-part ratio loses the third group entirely.

State the ratio in its simplest form

The three groups are in the ratio $5$ : $7$ : $8$.

Write down the two ratios

The two ratios share the letter b, which is the link between them.

Spot the problem with the link

b is worth 4 parts in the first ratio but only 2 parts in the second, so the ratios cannot be written side by side until b matches.

Find the lowest common multiple of the two values of b

4 is the smallest number that both 4 and 2 divide into, so make b worth 4 parts in both ratios.

Scale the first ratio

b is already 4 here, so the first ratio does not need changing.

Scale the second ratio

Multiplying both parts of the second ratio by 2 makes b worth 4 parts here too.

Check the link now matches

b is worth 4 parts in both scaled ratios, so they can be joined together.

Combine the ratios

Take a from the first ratio, the shared b, and c from the second ratio.

Check the combined ratio is in its simplest form

No number bigger than $1$ divides all three parts, so this is fully simplified.

Check the first ratio survives

Reading off the first two parts gives a : $b = 3$ : $4$, exactly as the question stated.

Check the second ratio survives

Reading off the last two parts and cancelling gives b : $c = 2$ : $7$, exactly as the question stated.

Reject the naive answer

Writing the numbers straight down would give b : $c = 4$ : 7, but the question says b : $c = 2$ : $7$. So this option is wrong.

Reject the next option

Here $a : b = 6 : 8 = 3 : 4$, which is right, but $b : c = 8 : 7$, which is not $2 : 7$. So c has not been scaled with b.

Reject the next option

Here a : $b = 3$ : $2$, but the question says a : $b = 3$ : $4$, so this option contradicts the first ratio.

Reject the last option

Here a : $b = 3$ : 8, not 3 : 4, so b has been scaled without scaling a. The first ratio has been broken.

State the answer

This is the only option that agrees with BOTH given ratios and is in its simplest form.