FN1–FN3, FN10, HN1–HN3, HN10 - Equivalent fractions and number order | Eduqas GCSE Mathematics

Use equal scaling to recognise and simplify equivalent fractions. You will convert between improper fractions and mixed numbers, locate signed fractions on a number line, and compare fractions using common denominators or exact decimals.

Equivalent fractions

Which is larger, $\frac35$ or $\frac58$ ? Comparing the top numbers alone will not answer that: the pieces have different sizes. Begin by seeing how the same amount can have different fraction names.

In $\frac35$ , the denominator $5$ says the whole is divided into five equal parts. The numerator $3$ says to take three of those parts. Split every fifth in half: there are now ten equal parts, and the same shaded amount contains six of them.

[DIAGRAM: asset_name: Equal amounts with different fraction names; asset_slug: equivalent-fraction-bars; caption: Three fifths and six tenths cover the same amount of the same whole.; description: Two aligned bars of equal length. The top is split into five equal parts with three shaded; the bottom into ten equal parts with six shaded. Labels show three fifths equals six tenths.]

Equivalent fractions

Equivalent fractions have the same value. Multiply or divide the numerator and denominator by the same non-zero number to preserve the value:

\frac35=\frac{3\times2}{5\times2}=\frac6{10}.

A denominator cannot be zero.

Why does this work? Multiplying the numerator and denominator by $2$ multiplies the fraction by $\frac22=1$ . Dividing both by a common factor reverses that scaling. This is cancellation, and it gives a fraction in its simplest form when numerator and denominator share no factor greater than $1$ .

Worked example 1: simplifying a fraction

Question. Write $\frac{18}{30}$ in its simplest form.

Plan. Find a common factor of $18$ and $30$ , then divide both by it.

\begin{aligned} \frac{18}{30}&=\frac{18\div6}{30\div6} &&\text{cancel the common factor }6\\ &=\frac35 \end{aligned}

The only positive factor shared by $3$ and $5$ is $1$ , so the simplification is complete.

Answer. $\frac35$ .

Check. Scaling back by $6$ gives $\frac{3\times6}{5\times6}=\frac{18}{30}$ .

You may cancel in stages instead: $\frac{18}{30}=\frac9{15}=\frac35$ . The next example adds a negative sign; cancellation changes the sizes of the integers, but preserves that sign.

Worked example 2: simplifying a negative fraction

Question. Write $-\frac{24}{42}$ in its simplest form.

Plan. Simplify the positive fraction, keeping the minus sign attached to the whole value.

\begin{aligned} -\frac{24}{42}&=-\frac{24\div6}{42\div6}\\ &=-\frac47 \end{aligned}

The numbers $4$ and $7$ have no common factor greater than $1$ .

Answer. $-\frac47$ .

Check. $-\frac{4\times6}{7\times6}=-\frac{24}{42}$ . The result is still negative and between $-1$ and $0$ .

Equivalent scaling is different from adding the same amount to both numbers.

Watch out

$\frac35\ne\frac{3+2}{5+2}=\frac57$ . Adding $2$ does not split every part in the same way. Multiply both numbers by $2$ to get the equal fraction $\frac6{10}$ .

Simplify a new signed fraction completely.

Mixed numbers

An improper fraction can contain whole units as well as part of another unit.

With a positive denominator, a proper fraction has a numerator smaller in size than the denominator; an improper fraction has a numerator at least as large in size. For instance, $\frac34$ is proper, $\frac{11}{4}$ is improper, and $\frac44=1$ is also improper.

Mixed number

A mixed number combines a whole number with a proper fraction. For example,

2\frac34=2+\frac34=\frac{11}{4}.

A minus sign in front applies to the whole mixed number: $-2\frac34=-\left(2+\frac34\right)=-\frac{11}{4}$ .

To turn an improper fraction into a mixed number, divide the numerator's size by the denominator. The quotient counts whole units; the remainder counts parts of the next unit. Keep the denominator because the size of each part has not changed.

Worked example 3: an improper fraction on a number line

Question. Write $\frac{11}{4}$ as a mixed number and state where it lies on a number line.

Plan. Group eleven quarters into whole units of four quarters.

\begin{aligned} 11&=2\times4+3 &&\text{two complete groups, three left}\\ \frac{11}{4}&=2\frac34 \end{aligned}

It is three quarters of the way from $2$ to $3$ , at the third quarter-step after $2$ .

Answer. $2\frac34$ , between $2$ and $3$ .

Check. Two whole units are eight quarters; eight quarters and three quarters give the original eleven quarters.

For the reverse conversion, count the parts in the whole units and add the remaining parts: multiply the whole-number size by the denominator, then add the numerator. The next example changes the sign and the direction of conversion.

Worked example 4: a negative mixed number

Question. Write $-2\frac34$ as an improper fraction and state where it lies on a number line.

Plan. Count the quarters in $2\frac34$ , then apply the minus sign to the whole amount.

\begin{aligned} 2\times4+3&=11 &&\text{quarters in the positive amount}\\ -2\frac34&=-\frac{11}{4} \end{aligned}

The value is $2\frac34$ units left of zero. It is between $-3$ and $-2$ , one quarter-step to the right of $-3$ .

Answer. $-\frac{11}{4}$ , between $-3$ and $-2$ .

Check. Its opposite, $\frac{11}{4}$ , is at the same distance to the right of zero; the two positions are reflections in zero.

Every interval between neighbouring integers represents one whole, so divide each into four equal steps when plotting quarters.

[DIAGRAM: asset_name: Positive and negative mixed numbers; asset_slug: mixed-number-line; caption: Eleven quarters and negative eleven quarters lie the same distance from zero.; description: One exact number line from minus three to three with quarter ticks. The points minus eleven quarters and eleven quarters are marked at minus two and three quarters and two and three quarters, respectively; zero and all integer ticks are labelled.]
Eleven quarters and negative eleven quarters lie the same distance from zero.

The quotient-and-remainder method also handles exact whole numbers: $\frac{12}{4}=3$ , with no fractional part left.

Watch out

$-2\frac34$ means $-2-\frac34$ , not $-2+\frac34$ . Apply the minus sign to both the whole units and the fractional part.

Convert a new negative mixed number and locate its value.

Ordering fractions

Compare fractions by expressing them in equal-sized parts, or by using exact decimal values.

When positive denominators match, the larger numerator gives the larger value: $\frac7{12}>\frac5{12}$ . The pieces are the same size, and seven pieces exceed five. With different denominators, first scale to a common denominator. Any common multiple works; choosing a small one keeps the numbers manageable.

Worked example 5: comparing with a common denominator

Question. Which is larger, $\frac35$ or $\frac58$ ? Show your working.

Plan. Rewrite both fractions in fortieths, so their numerators count equal-sized parts.

\begin{aligned} \frac35&=\frac{3\times8}{5\times8}=\frac{24}{40}\\ \frac58&=\frac{5\times5}{8\times5}=\frac{25}{40}\\ \frac{24}{40}&<\frac{25}{40} \end{aligned}

Answer. $\frac58$ is larger.

Check. An exact decimal comparison agrees: $\frac35=\frac6{10}=0.6$ , while $\frac58=\frac{625}{1000}=0.625$ , which is larger than $0.600$ .

To use an exact terminating decimal, scale the denominator to $10$ , $100$ , $1000$ or another power of ten. For example, $\frac7{20}=\frac{35}{100}=0.35$ . This works particularly well for halves, quarters, fifths, eighths and tenths. If the decimal would need rounding, a common denominator keeps the comparison exact.

Negative fractions follow the number-line rule from the first lesson. Among two negative numbers, the one closer to zero is greater. Keep each minus sign attached when comparing signed numerators.

Worked example 6: changing both fractions to negative values

Question. Write $-\frac35$ and $-\frac58$ in increasing order, using $<$ .

Plan. Use fortieths again, then compare the signed numerators.

\begin{aligned} -\frac35&=-\frac{24}{40}\\ -\frac58&=-\frac{25}{40}\\ -25&<-24 &&\text{further left on a number line} \end{aligned}

Answer. $-\frac58<-\frac35$ .

Check. The exact decimals are $-0.625$ and $-0.600$ . The first is further left, so it is smaller.

A positive fraction exceeds a negative fraction without any conversion. Equivalent fractions have equal positions: $\frac35=\frac6{10}$ , so statements such as $\frac35\le\frac6{10}$ are true because the symbol allows equality.

Watch out

For negatives, the larger positive size gives the smaller value. Since $\frac58>\frac35$ , their negatives satisfy $-\frac58<-\frac35$ . Do not discard the signs after finding a common denominator.

The opening comparison is now justified: $\frac58$ exceeds $\frac35$ by one fortieth. The numerators could only be compared directly after the denominators matched.

Quick Recap

Multiply or divide numerator and denominator by the same non-zero number to preserve a fraction's value.
In simplest form, numerator and denominator have no common factor greater than $1$ .
Use whole groups and a remainder to convert between improper fractions and mixed numbers; a leading minus sign applies to the entire mixed number.
Compare fractions using common positive denominators or exact decimals, keeping their signs.

$\frac58>\frac35$ , because twenty-five fortieths exceed twenty-four fortieths.

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FN1–FN3, FN10, HN1–HN3, HN10 - Equivalent fractions and number order

Equivalent fractions

Mixed numbers

Ordering fractions

Quick Recap

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