Hard GCSE FDP equivalence Questions

Percent means 'out of $100$'

A percentage is a number out of $100$, so write it as a fraction over $100$.

Clear the decimal from the fraction

Multiply top and bottom by $10$ so there is no decimal left in the fraction.

Find the highest common factor

The biggest number dividing both $875$ and $1000$ is $125$.

Divide top and bottom by the HCF

Dividing top and bottom by the same number simplifies without changing the value.

Check by converting back

Turning the fraction back into a percentage returns the value we started with.

Check the size makes sense

The percentage is bigger than a half ($0.5$) of the whole, which the fraction agrees with.

Write it as a decimal too

The same amount as a decimal is the percentage divided by $100$.

Why over $100$

Because 'per cent' literally means 'per hundred', every percentage starts life over $100$.

Watch a common slip

The denominator is $100$, not $10$; per cent means per hundred.

Cancel in stages if it helps

You can cancel with smaller factors step by step instead of finding the HCF at once.

Say it in words

Reading the fraction aloud keeps its meaning clear.

Real-world use

Sale signs and statistics often need turning from a percentage into a fraction.

Exam tip: simplest form

Always cancel fully; the simplest form is usually where the mark sits.

Double-check the cancelling

Confirm the top and bottom share no further common factor.

State the simplest form

So $87.5\% = \frac{7}{8}$ in its simplest form.

Read the problem

Rent is $30\%$, food is one quarter and travel is $0.2$ of the income.

Turn every share into the same form

To combine the shares we first write each one as a fraction of the whole.

Convert the rent share

Rent is $\frac{3}{10}$ of the whole.

Convert the food share

Food is $\frac{1}{4}$ of the whole.

Convert the travel share

Travel is $\frac{1}{5}$ of the whole.

Add the known shares

Add the parts that are already accounted for.

Subtract from the whole

Everything left over is one whole minus the shares already used.

Check the total is one whole

All the shares including the leftover must add back up to one whole.

Check with decimals

Doing the same sum in decimals gives a matching answer.

Check the size makes sense

The leftover is smaller than a half ($0.5$), which feels reasonable for what is described.

Watch a common slip

Subtract only once all the given shares are added, or errors creep in.

As a percentage

The same leftover as a percentage is found by multiplying by $100$.

Why a common denominator helps

Fractions can only be added when they are cut into the same-size pieces.

Say it in words

Reading the answer aloud keeps its meaning clear.

State the fraction left

So $\frac{1}{4}$ of the whole is left, in its simplest form.

Percent means 'out of $100$'

To change a fraction into a percentage, multiply it by $100$.

Multiply the fraction by $100$

The division does not come out whole; it leaves a repeating third, so we write it exactly as $16\frac{2}{3}$.

Attach the percent sign

Writing the percent sign shows this is a number out of $100$.

Check with a benchmark

The fraction is smaller than a half ($0.5$), and the percentage is on the matching side of $50\%$.

Why multiplying by $100$ works

One whole is $100\%$, so a fraction of a whole is that same fraction of $100\%$.

Check by converting back

Turning the percentage back into a fraction returns the fraction we started with.

Watch a common slip

Do not just push the two numbers together; you must multiply by $100$.

Say it in words

Reading the answer aloud as 'per cent' (per hundred) helps it stick.

Sense check the range

A part of one whole should land between $0\%$ and $100\%$, and it does.

Note the recurring decimal

Because the denominator has a factor of $3$, the decimal recurs, so we keep an exact fraction inside the percentage.

Real-world use

Percentages like this appear on test scores, discounts and statistics all the time.

Link to a fraction you know

You can also reach this by scaling a simpler fraction, such as a half, a quarter or a third.

Exam tip: keep it exact

When the percentage is not a whole number, write the exact fraction rather than rounding.

Reflect on the method

Multiplying by $100$ is the one move that turns any fraction into a percentage.

State the percentage

So $\frac{1}{6} = 16\frac{2}{3}\%$.

Put everything in the same form

Fractions, decimals and percentages are easiest to order when all written as decimals.

Convert $0.44$ to a decimal

Writing the decimal as a decimal gives $0.44$.

Convert $\frac{2}{5}$ to a decimal

Writing the fraction as a decimal gives $0.4$.

Convert $43\%$ to a decimal

Writing the percentage as a decimal gives $0.43$.

Convert $\frac{9}{20}$ to a decimal

Writing the fraction as a decimal gives $0.45$.

Compare the decimals

Line the decimals up and read them off from smallest to largest.

Write the answer in the original forms

Swap each decimal back for the fraction, decimal or percentage the question used.

Double-check the ends

Check the first and last entries really are the extreme values.

Watch a common slip

Comparing a raw percentage against a decimal without converting reverses the order.

Sense check with a number line

Imagine the values on a number line: the order should read straight across.

Reflect

Converting to a single form removes the guesswork when ordering mixed numbers.

Line them up neatly

Presenting the final list clearly earns the communication marks.

Spot-check one conversion

Re-checking a single conversion guards against a slip.

Watch equal-looking values

Extra zeros after the point do not change a value; look at the leading digits.

State the ordered list

In smallest to largest order: $\frac{2}{5}, 43\%, 0.44, \frac{9}{20}$.

Read the problem

The coat costs three quarters of the price, so the saving is the rest.

Decide what to find

The question asks for a percentage, so we head towards a number out of $100$.

Write down the given part

First note the fraction as a fraction of the whole.

Find the part we actually want

The part asked for is what is left after taking away the given part.

Multiply by $100$

Multiplying the fraction by $100$ turns it into a percentage.

Check by converting back

Turning the percentage back into a fraction returns the same part.

Check the whole adds to $100\%$

The two parts should together make the full $100\%$.

Check the size makes sense

The part is smaller than a half ($0.5$), which fits the situation.

Watch a common slip

Read carefully whether the question wants the given part or the leftover part.

Show it as a decimal too

The same percentage as a decimal is found by dividing by $100$.

Why multiply by $100$

One whole is $100\%$, so a fraction of the whole is that fraction of $100\%$.

Say it in words

Stating the answer in words keeps the meaning clear for the reader.

Real-world use

Splitting a group into percentages is exactly how survey results are reported.

Reflect

Fractions, decimals and percentages are three views of one amount.

State the percentage

So the answer is $25\%$.