N10, N12 - Fractions, decimals and percentage operators

Fractions, decimals and percentages are different ways to describe the same proportion. N10 connects terminating decimals with exact fractions; N12 uses fractions and percentages as operators, both to find a part and to recover its whole.

Terminating decimals as exact fractions

A jug contains $0.375$ litres, and this is $\frac35$ of its capacity. Can you find the full capacity without rounding? First turn the decimal into an exact fraction.

You already know

A fraction represents division. Multiplying or dividing its numerator and denominator by the same non-zero number preserves its value.

A terminating decimal ends after finitely many decimal places. Its final place tells you the denominator: tenths use ten, hundredths use a hundred, and so on. Leading zeros matter: $0.04=\frac4{100}=\frac1{25}$ , not $\frac4{10}$ .

Worked example 1: decimal to fraction and back

Question. Write $0.375$ as a fraction in simplest form, then check by conversion back.

Plan. Use the thousandths column and simplify.

\begin{aligned} 0.375&=\frac{375}{1000} && \text{three decimal places} \\ &=\frac38 && \text{divide both by }125 \\ \frac38&=\frac{3\times125}{8\times125}=0.375 && \text{make thousandths again} \end{aligned}

Why? The denominator counts the size of each place, rather than the number of non-zero digits.

Answer. $\frac38$ .

Check. $3\div8=0.375$ .

Change only the whole-number part, from zero to one. The fraction will now be improper.

Worked example 2: a value above one

Question. Write $1.375$ as a fraction in simplest form and check it.

Plan. Keep all the digits over a thousand before simplifying.

\begin{aligned} 1.375&=\frac{1375}{1000} && \text{read thousandths} \\ &=\frac{11}{8} && \text{divide both by }125 \\ \frac{11}{8}&=1+\frac38=1.375 && \text{separate the whole} \end{aligned}

Why? A decimal above one needs a numerator larger than its denominator.

Answer. $\frac{11}{8}$ , or $1\frac38$ .

Check. The fraction exceeds Example 1 by exactly one.

Signs and trailing zeros behave normally: $3.5=\frac72$ , $-1.28=-\frac{32}{25}$ and $0.5=\frac5{10}=\frac{50}{100}$ . To convert in the other direction, use division or an equivalent power-of-ten denominator: $\frac7{20}=\frac{35}{100}=0.35$ .

Here is a useful surprise: in simplest form, a fraction terminates exactly when its positive denominator contains only prime factors two and five. Powers of ten contain only those primes. For instance, $\frac9{40}=\frac{225}{1000}=0.225$ , while a remaining factor three prevents a power-of-ten denominator. Simplify first: $\frac3{12}=\frac14$ terminates.

Watch out

Keep the sign and count every decimal place. Wrong: $-0.04=\frac4{10}$ . Right: $-0.04=-\frac1{25}$ .

Change only the whole-number part again.

Percentages and equivalent proportions

Per cent means “per hundred”. So $35\%=\frac{35}{100}=\frac7{20}=0.35$ ; the percentage symbol is part of the value.

Percentage

A percentage describes an amount per hundred of a reference whole. $100\%$ is one whole; $125\%$ is $1.25$ wholes.

Divide the percentage number by a hundred to find its decimal. Reverse this by multiplying a decimal by a hundred and attaching the percentage sign. Fractions can go through decimals too: $\frac38=0.375=37.5\%$ .

Worked example 3: equivalent forms of a percentage

Question. Write $0.6\%$ as a decimal and a fraction in simplest form.

Plan. Divide by a hundred, then express the decimal as thousandths.

\begin{aligned} 0.6\%&=0.6\div100=0.006 && \text{per hundred} \\ &=\frac6{1000}=\frac3{500} && \text{simplify} \end{aligned}

Why? Less than one per cent must be less than $0.01$ of the whole.

Answer. $0.006$ and $\frac3{500}$ .

Check. $0.006\times100=0.6$ , giving $0.6\%$ again.

Change only the percentage to $160\%$ . The same conversion now produces more than one whole.

Worked example 4: more than a whole

Question. Write $160\%$ as a decimal and a fraction in simplest form.

Plan. Divide by a hundred and simplify the resulting fraction.

\begin{aligned} 160\%&=160\div100=1.6 && \text{convert to a decimal} \\ &=\frac{16}{10}=\frac85 && \text{simplify} \end{aligned}

Why? A percentage above a hundred represents more than the reference whole.

Answer. $1.6$ and $\frac85$ .

Check. $\frac85=1+\frac35=1.6$ .

The reference matters. A container can have $125\%$ of another container's capacity, but a subset cannot contain $125\%$ of the members of its own whole. Common equivalents include $\frac12=0.5=50\%$ , $\frac1{20}=0.05=5\%$ , $\frac54=1.25=125\%$ and $\frac3{200}=0.015=1.5\%$ .

To compare proportions, use a common form: $58\%=0.58<\frac35=0.6<0.62$ . Comparing the printed numerals alone ignores their meaning.

Watch out

Wrong: $0.6\%=0.6$ . Right: $0.6\%=0.006$ . A percentage number must be divided by a hundred.

Change only the percentage in the second example.

Taking a fraction and recovering the whole

An operator acts on a quantity. Taking $\frac35$ of $40$ means splitting forty into five equal parts, then taking three: $40\div5\times3=24$ .

Fraction operator

Taking $\frac ab$ of a quantity means multiplying the quantity by $\frac ab$ , with $b\ne0$ . “Of” asks for multiplication here.

To recover the whole, undo that operation by dividing by the fraction, or multiplying by its reciprocal.

[DIAGRAM: asset_name: A fraction operator and its inverse; asset_slug: 1ma1_m08_fraction_operator; caption: Multiplying by three fifths takes 40 to 24; multiplying by five thirds recovers 40.; description: A left box labelled whole 40 and a right box labelled part 24 are linked by a forward arrow labelled multiply by 3 over 5 and a return arrow labelled divide by 3 over 5, equivalent to multiply by 5 over 3.]
Multiplying by three fifths takes 40 to 24

Worked example 5: finding a part and undoing it

Question. Find $\frac34$ of $2.8\,\mathrm{m}$ . Show how this part recovers the original length.

Plan. Divide by four and multiply by three; reverse those steps to recover the whole.

\begin{aligned} \text{part}&=2.8\div4\times3=2.1\,\mathrm{m} && \text{three equal quarters} \\ \text{whole}&=2.1\div3\times4=2.8\,\mathrm{m} && \text{undo the operator} \end{aligned}

Why? The known part contains three quarters, so divide it by three to find one quarter.

Answer. The part is $2.1\,\mathrm{m}$ ; its whole is $2.8\,\mathrm{m}$ .

Check. $\frac34\times2.8=2.1$ , with the same unit.

Change only the fraction to $\frac54$ . This time the operator enlarges the quantity.

Worked example 6: an improper fraction operator

Question. Find $\frac54$ of $2.8\,\mathrm{m}$ and recover the original length from it.

Plan. Take five quarters, then divide by that same fraction.

\begin{aligned} \text{part}&=2.8\div4\times5=3.5\,\mathrm{m} && \text{five quarters} \\ \text{whole}&=3.5\div5\times4=2.8\,\mathrm{m} && \text{reverse the steps} \end{aligned}

Why? “Part” need not mean smaller when the fraction exceeds one.

Answer. $3.5\,\mathrm{m}$ , recovering $2.8\,\mathrm{m}$ .

Check. $3.5\times\frac45=2.8$ .

If $\frac78$ of a tank's capacity is $35$ litres, one eighth is $35\div7=5$ litres, so capacity is $5\times8=40$ litres. For the opening jug, $\frac38\div\frac35=\frac38\times\frac53=\frac58=0.625$ litres exactly.

Watch out

If $\frac35$ of a whole is $24$ , multiplying $24$ by $\frac35$ repeats the operation. Recover the whole with $24\div\frac35=40$ .

Change only the length in the improper-fraction example.

Percentages as multipliers

A percentage operator is the same idea with a fraction out of a hundred. For $18\%$ of $250$ , calculate $0.18\times250=45$ . Alternatively add $10\%$ , $5\%$ and $3\%$ : $25+12.5+7.5=45$ .

Percentage operator

p\%\text{ of }Q=\frac p{100}\times Q

Use the stated reference quantity $Q$ . To recover $Q$ from a non-zero percentage part, divide by $p/100$ .

Not on the formula sheet: learn it.

If $35\%$ of eighty counters are removed, the remaining $65\%$ is $0.65\times80=52$ . The removed $28$ and remaining $52$ add to the original eighty. Above a hundred per cent works too: $140\%$ of $50$ is $1.4\times50=70$ .

Worked example 7: finding and reversing a percentage

Question. Find $5\%$ of $600$ , then recover the whole from that part.

Plan. Convert the percentage to a multiplier and reverse by division.

\begin{aligned} \text{part}&=0.05\times600=30 && \text{five per hundred} \\ \text{whole}&=30\div0.05=600 && \text{undo multiplication} \end{aligned}

Why? The percentage always acts on the reference whole, not automatically on the last number written.

Answer. The part is $30$ ; its whole is $600$ .

Check. Ten per cent is sixty, so five per cent is thirty.

Change only the percentage from $5\%$ to $0.5\%$ . The part should be ten times smaller.

Worked example 8: less than one per cent

Question. Find $0.5\%$ of $600$ , then recover the whole from that part.

Plan. Divide the percentage number by a hundred before multiplying.

\begin{aligned} 0.5\%&=0.005 && \text{convert the percentage} \\ \text{part}&=0.005\times600=3 && \text{apply the operator} \\ \text{whole}&=3\div0.005=600 && \text{reverse it} \end{aligned}

Why? $0.5\%$ is one half of one per cent, not one half of the whole.

Answer. The part is $3$ ; its whole is $600$ .

Check. One per cent of six hundred is six; half of six is three.

Recovery works for any stated non-zero percentage: if $40\%$ of a mass is $14\,\mathrm{kg}$ , the whole is $14\div0.4=35\,\mathrm{kg}$ . Check $0.4\times35=14$ . A percentage alone cannot tell you an amount: $20\%$ of fifty and of two hundred are different quantities.

Watch out

Wrong: if $40\%$ of a mass is $14\,\mathrm{kg}$ , the whole is $0.4\times14$ . This applies the operator twice. Right: undo it with $14\div0.4=35\,\mathrm{kg}$ .

Change only the whole in Example 8.

Quick Recap

Place value converts terminating decimals to exact fractions; simplify the result.
Per cent means per hundred, including percentages below one or above a hundred.
Multiply by a fraction or percentage multiplier to find its part.
Divide by the same non-zero operator to recover the reference whole.

The opening jug has capacity exactly $\frac58$ litres, or $0.625$ litres.

Last Accessed

Papers 1, 2 and 3

N10, N12 - Fractions, decimals and percentage operators

Terminating decimals as exact fractions

Percentages and equivalent proportions

Taking a fraction and recovering the whole

Percentages as multipliers

Quick Recap

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