Hard GCSE Fractions of amounts Questions

Plan the stages

Each fraction is spent from the money remaining at that point. Work stage by stage to the amount left.

charity: spend the fraction

$1$ ninths of $2160$ is $240$ for charity.

Money left after charity

$1920$ remains after paying charity.

rent: spend the fraction

$5$ twelfths of $1920$ is $800$ for rent.

Money left after rent

$1120$ remains after paying rent.

food: spend the fraction

$1$ quarters of $1120$ is $280$ for food.

Money left after food

$840$ remains after paying food.

State the final amount left

After all three, $840$ is left.

Sense check

The three amounts plus what is left add back to $2160$.

Check the amount after the first stage

After the first spend $1920$ remained, which the next fraction used.

Watch out for the common error

Only the first fraction is of the full amount; the later ones are of what is left.

Interpret the last stage

The final spend was a fraction of the $1120$ left before it.

Alternative for the last stage

Keeping the other fraction of the $1120$ gives the $840$ left directly.

Note the answer is exact

Every step used exact whole numbers or fractions, so there is no rounding — the answer is exact.

State the answer

The final answer is $840$ pounds.

Understand the problem

$15$ red counters are $3$ eighths of the total. Find the total, then remove some blue.

Find one part

$15$ is $3$ eighths, so one eighth is $5$.

Find the total

The total is $8$ of those parts: $40$ counters.

Find the blue counters

Taking the red from the total leaves $25$ blue.

Start removing $2/5$ of the blue

$2$ fifths of the blue are removed; divide by $5$ first.

Find how many are removed

$10$ blue counters are removed.

Find the blue left

That leaves $15$ blue.

Find the total left in the bag

The red are untouched, so add them to the blue left: $30$.

Check by removing from the total

Removing $10$ from the whole $40$ also gives $30$.

Sense check

$3$ eighths of $40$ is $15$ red, confirming the total.

Interpret the removal fraction

The removal was a fraction of the blue only.

Watch out for the common error

Only the blue are reduced; the red stay the same.

Note the answer is exact

Every step used exact whole numbers or fractions, so there is no rounding — the answer is exact.

Reflect on the method

Find the total, work out what leaves, subtract once.

State the answer

The final answer is $30$ counters.

Understand the problem

Two different fractions of the same number $n$ are subtracted to give a known result. Combine them first.

Subtract the fractions

Using the common denominator $12$, the difference of the two fractions is $\frac{7}{12}$.

Rewrite the equation

So $\frac{7}{12}$ of the number is $42$.

Find one part

$42$ is $7$ twelfths, so divide by $7$.

Find the number

$n$ is $12$ of those parts: $72$.

Find the first fraction of $n$

$5$ sixths of $72$ is $60$.

Find the second fraction of $n$

$1$ quarters of $72$ is $18$.

Check the difference

The two amounts differ by $42$, exactly as required.

Sense check

Adding the result to the smaller amount gives the larger one.

Alternative: divide by the fraction

Dividing by $\frac{7}{12}$ (multiply by its reciprocal) also gives $72$.

Watch out for the common error

Do not work out each fraction of a guessed number; combine the fractions first.

Why $n$ is larger than the result

Because $\frac{7}{12}$ is less than $1$, $n$ must be bigger than $42$.

Note the answer is exact

Every step used exact whole numbers or fractions, so there is no rounding — the answer is exact.

Reflect on the method

Combining like fractions of $n$ turns two unknowns into one simple reverse step.

State the answer

The final answer is $72$.

Plan the calculation

Each fraction is of the same amount $900$. Find each part, add them, then subtract from $900$ to get the car travellers group.

Find the walkers

$2$ fifths of $900$ is $360$ (walkers).

Find the bus riders

$1$ thirds of $900$ is $300$ (bus riders).

Find the cyclists

$1$ sixths of $900$ is $150$ (cyclists).

Add the amounts used

These groups total $810$.

Add the fractions used

As one fraction, $\frac{9}{10}$ of the total is accounted for.

Find the fraction left

The car travellers group is $\frac{1}{10}$ of the total.

Find the car travellers group

Subtract the others from $900$ to leave $90$.

Interpret the car travellers group

So the car travellers group comes to $90$.

Check with the fraction left

$\frac{1}{10}$ of $900$ is $90$, matching the subtraction.

Sense check

All the groups add back to $900$.

Watch out for the common error

Each fraction is of the original $900$, so add them before subtracting once.

Note the answer is exact

Every step used exact whole numbers or fractions, so there is no rounding — the answer is exact.

Reflect on the method

Add up everything accounted for, then one subtraction gives the last group.

State the answer

The final answer is $90$.

Plan: work backwards

The $192$ is the price after the reduction, not the original. Find what fraction of the original it is.

The reduction as a fraction

The phone is reduced by $1$ fifths of its original price.

Fraction of the price remaining

After taking off $1$ fifths, the sale price is $\frac{4}{5}$ of the original.

Link the fraction to the sale price

So $192$ is $4$ fifths of the original price.

Find one part

Divide by $4$ to find one fifth of the original.

Find the original price

The original is $5$ of those parts: $240$.

Check the reduction

The amount taken off is $48$.

Check the sale price

Taking off $48$ leaves $192$, exactly the sale price.

Alternative one-line method

Dividing the sale price by $4$ and multiplying by $5$ reaches $240$ directly.

Watch out for the common error

Do not take the reduction fraction of the sale price — the $192$ is already reduced.

Why the original is larger

The sale price is only $\frac{4}{5}$ of the original, so the original must be bigger, which it is.

Sense check

The reduction and the sale price add back to the original.

Note the answer is exact

Every step used exact whole numbers or fractions, so there is no rounding — the answer is exact.

Reflect on the method

The key idea: the given price is a known fraction of the unknown original.

State the answer

The final answer is $240$ pounds.