C1.4; E1.4 - Equivalent fractions and mixed numbers

Fraction notation

Split one metre into four equal parts. Each part is one quarter of a metre, written $\frac14\,\mathrm{m}$. Three such parts have length $\frac34\,\mathrm{m}$. The parts must be equal and must refer to the same-sized whole.

Fractions can describe more than one whole. After four quarter-metres fill one metre, three more quarter-metres give a total of $\frac74\,\mathrm{m}$. You can instead count one whole metre and three quarters: $1\frac34\,\mathrm{m}$.

[DIAGRAM: asset_name: Seven quarters as a mixed number; asset_slug: seven_quarters; caption: Seven quarter-metres fill one whole metre and three quarters of another.; description: Two equal one-metre bars are each divided into four equal parts. All four parts of the first bar and three of the second bar are shaded blue. The remaining quarter is unshaded. Labels show seven quarters equals one and three quarters, with each complete bar labelled one metre.]

The denominator gives the unit being counted. Seven quarters are $\frac74$, not $\frac47$: changing the order would count four sevenths instead. An improper fraction is valid notation; the word “improper” does not mean the value is wrong. For example, $\frac{12}{4}=3$ represents exactly three wholes.

Choose the classification by comparing the numerator and denominator.

Simplest form

Two quarters and one half cover the same part of the same whole. The number of pieces changes when you divide each half into two, but the total amount does not. This is why multiplying both parts of a fraction by the same non-zero number preserves its value.

For example, $\frac25=\frac{2\times3}{5\times3}=\frac6{15}$: each fifth has been split into three fifteenths. To simplify, reverse the process. Dividing by the HCF removes all common factors in one step; several smaller common-factor divisions also work.

The next fraction is larger in numerator and denominator, but reduces to the same amount. A first common factor need not be the highest: you can simplify in stages if you check again at the end.

Keep an answer requested as a fraction in integer numerator and denominator form, and simplify fully. For a negative fraction, the sign stays: $-\frac68=-\frac34$. It is convenient to keep the denominator positive.

A final common-factor check tells you when to stop.

Mixed-number conversions

Changing between an improper fraction and a mixed number changes the way you count, rather than the amount. Count complete groups of the denominator to find wholes. Count all the fractional pieces to reverse the conversion.

For an improper fraction, divide the numerator by the denominator: the quotient counts whole groups and the remainder stays over the same denominator. For a mixed number, each whole contributes one denominator's worth of pieces; add the fractional pieces afterwards. These conversion rules are knowledge to learn rather than supplied formulas.

Now reverse the direction: the whole groups are given, and you need a single count of fractional pieces.

When a mixed number has a leading minus sign, it applies to the whole value: $-2\frac13=-(2+\frac13)=-\frac73$, rather than $-2+\frac13$. Convert the positive size, then keep the minus sign in front.

Use the quotient and remainder to check both directions.

Quick Recap

• The numerator counts pieces; the denominator names the equal-sized pieces.
• Equivalent fractions have the same value; multiply or divide both parts together.
• Simplest form has no common factor greater than 1 left.
• Divide to find wholes and a remainder; multiply the wholes by the denominator to convert back.

Seven quarter-metres are $\frac74\,\mathrm{m}=1\frac34\,\mathrm{m}$, answering the opening question.