N11, R3, R4, R5, R6, R7, R8 - Ratios, fractions and proportional sharing

Ratios describe relative amounts rather than fixed totals. You will connect ratios to fractions and linear relationships, share quantities, combine linked ratios and track changes to mixtures. These methods help you keep a recipe's flavour while changing its size.

Ratio notation and multiplicative relationships

Can two drinks with different amounts of concentrate taste equally strong? A drink with concentrate to water $2:3$ and one with $4:6$ have the same composition: the second doubles both ingredients. Adding one part of each instead gives $3:4$ , which changes the flavour.

You already know

Equivalent fractions have equal values. Multiplying the numerator and denominator by the same non-zero number preserves a fraction's value; the same idea preserves a ratio.

The order of the names sets the order of the numbers. Concentrate to water $2:3$ means two equal-sized parts of concentrate for every three of water. It says nothing about the size of each part.

Equivalent ratios

Equivalent ratios compare the same quantities in the same order and are made by scaling every term by the same non-zero factor. For positive amounts use a positive factor. For example, $2:3=4:6$ , whereas $3:4$ is different.

The ratio also gives a multiplier: water is $\frac32$ times concentrate, and concentrate is $\frac23$ times water. To simplify comparable measurements, use the same unit first. To clear decimals, scale all terms by a power of ten: $0.6:1.5=6:15=2:5$ . For three terms, scale all three: $8:12:20=2:3:5$ . These ratio rules are not on the formula sheet: learn them.

Worked example 1: a ratio with equal units

Question. Simplify $1.5\,\mathrm m:60\,\mathrm{cm}$ .

Plan. Convert metres to centimetres, then divide both terms by their highest common factor.

\begin{aligned} 1.5\,\mathrm m:60\,\mathrm{cm}&=150:60 && \text{use centimetres for both}\\ &=5:2 && \text{divide both terms by }30 \end{aligned}

Why this step? A ratio of lengths compares equal units, so $1.5:60$ would compare the wrong amounts.

Answer. $5:2$ .

Check. $5\times30=150$ and $2\times30=60$ .

The second example changes the form of the given terms to fractions. Equal scaling still does the work.

Worked example 2: fractional terms and a leading one

Question. Express $\frac12:\frac34$ in simplest whole-number form and then in the form $1:n$ .

Plan. Clear both denominators, then divide both terms by the first term.

\begin{aligned} \frac12:\frac34&=2:3 && \text{multiply both by }4\\ 2:3&=1:\frac32 && \text{divide both by }2 \end{aligned}

Why this step? The form $1:n$ requires the first term to be one; it does not require the second to be a whole number.

Answer. $2:3$ , then $1:\frac32$ .

Check. Both ratios have first term divided by second equal to $\frac23$ .

A ratio has to preserve a multiplier, rather than a difference.

Watch out

Wrong: $2:3=3:4$ , because one was added to both terms. The fractions $\frac23$ and $\frac34$ differ. Correct: multiply both terms by the same factor, for example $2:3=4:6$ .

Try the fractional version with a different first term.

Part-to-part and part-to-whole fractions

The denominator of a fraction tells you what you are comparing with, so name it before calculating.

For blue to green $2:3$ , blue is $\frac23$ of green but $\frac25$ of the whole. Green is $\frac32$ of blue: a comparison fraction can exceed one.

[DIAGRAM: asset_name: Part-to-part and part-to-whole ratios; asset_slug: 1ma1_m36_ratio_parts; caption: Two blue parts and three green parts make five parts altogether.; description: Five equal rectangles, the first two blue and the final three green, distinguish blue to green 2 to 3 from blue to total 2 to 5.]

One quantity as a fraction of another

Divide the first quantity by the comparison quantity, using common units. A part-to-whole fraction uses the complete total as its denominator; a part-to-part fraction uses the named other part.

If a collection contains only red and white beads and $\frac37$ are red, the remaining $\frac47$ are white. Thus red to whole is $3:7$ and red to white is $3:4$ . You need the word “only”: if there are other colours, their split cannot be inferred.

Worked example 3: choosing the denominator

Question. Two ribbons are $18\,\mathrm{cm}$ and $30\,\mathrm{cm}$ long. Express the shorter as a fraction of the longer and of the total.

Plan. Use the named comparison length as each denominator.

\begin{aligned} \text{shorter of longer}&=\frac{18}{30}=\frac35 && \text{compare with longer}\\ \text{total}&=18+30=48\,\mathrm{cm} && \text{combine both lengths}\\ \text{shorter of total}&=\frac{18}{48}=\frac38 && \text{compare with whole} \end{aligned}

Why this step? The same numerator can give different fractions when the comparison changes.

Answer. $\frac35$ of the longer and $\frac38$ of the total.

Check. $\frac38\times48=18$ .

Now change just the direction of the comparison: the longer is the numerator.

Worked example 4: a fraction greater than one

Question. For the same ribbons, express the longer as a fraction of the shorter and of the total.

Plan. Keep the denominator meanings, but use the longer length as numerator.

\begin{aligned} \text{longer of shorter}&=\frac{30}{18}=\frac53 && \text{compare the two parts}\\ \text{longer of total}&=\frac{30}{48}=\frac58 && \text{compare with the whole} \end{aligned}

Why this step? $30$ exceeds $18$ , so this first comparison must be greater than one.

Answer. $\frac53$ and $\frac58$ .

Check. The two whole-fractions add to $\frac38+\frac58=1$ .

Keep the denominator's label beside your working until the choice is secure.

Watch out

Wrong: blue to green $2:3$ means $\frac23$ of all counters are blue. The 3 counts green parts, not all parts. Correct: the whole has $2+3=5$ parts, so blue is $\frac25$ of the whole.

Change the ribbon lengths and make the two comparisons again.

Sharing quantities and recovering totals

Find the size of one equal part by matching the known amount to the correct number of ratio parts.

A total matches the sum of the terms; a difference matches their difference; a known share matches that share's term. For flour to sugar $5:2$ , $140\,\mathrm g$ of sugar gives one part $140\div2=70\,\mathrm g$ , flour $350\,\mathrm g$ and total $490\,\mathrm g$ .

Worked example 5: sharing a total

Question. Divide £150 between A and B in the ratio $2:3$ .

Plan. Find one of the five total parts, then multiply for each share.

\begin{aligned} \text{total parts}&=2+3=5 && \text{add both shares}\\ \text{one part}&=150\div5=30 && \text{divide the total}\\ A&=2\times30=60 && \text{take two parts}\\ B&=3\times30=90 && \text{take three parts} \end{aligned}

Why this step? Both shares use the same £30 part.

Answer. A gets £60 and B gets £90.

Check. $60+90=150$ and $60:90=2:3$ .

Change only what the given £150 represents: it is now the difference, rather than the total.

Worked example 6: recovering a total from a difference

Question. A and B share money in the ratio $2:3$ . B gets £150 more than A. Find the total.

Plan. Match the difference to one part, then find all five parts.

\begin{aligned} \text{difference in parts}&=3-2=1 && \text{compare the shares}\\ \text{one part}&=150\div1=150 && \text{use the known difference}\\ \text{total}&=5\times150=750 && \text{add five equal parts} \end{aligned}

Why this step? The given money belongs to the gap between the shares.

Answer. £750.

Check. The shares £300 and £450 differ by £150 and total £750.

Part-to-whole language needs the same care: first share to total $2:5$ means first to second $2:3$ , since three of the five parts remain.

Watch out

Wrong: for a $2:3$ split with a £150 difference, divide £150 by $2+3$ . That treats the difference as the total. Correct: divide by $3-2=1$ , then multiply by five for the total.

The next difference changes the size of one part.

Equal proportions and linear relationships

Equal ratios describe the same multiplicative relationship, even when their numbers look different.

Proportion

Proportion is equality of ratios. For example, $a:b=3:5$ means $\frac ab=\frac35$ , with $b\ne0$ . The pair $(12,20)$ fits; $(12,18)$ does not.

Multiplying $\frac ab=\frac35$ by $5b$ gives $5a=3b$ ; dividing by 5 gives $a=\frac35b$ . Cross-products and equal scaling describe the same relationship. This equality rule is not on the formula sheet: learn it.

Worked example 7: equal scaling

Question. Find $x$ if $\frac{x}{15}=\frac45$ .

Plan. Scale the denominator from 5 to 15, and scale the numerator equally.

\begin{aligned} \text{scale factor}&=15\div5=3 && \text{match denominators}\\ x&=4\times3=12 && \text{scale the numerator} \end{aligned}

Why this step? Equal fractions require equal scaling of both terms.

Answer. $x=12$ .

Check. $\frac{12}{15}=\frac45$ ; equivalently $5x=4\times15=60$ .

Change the numerical pair into variable quantities. The same multiplier now describes every matching pair.

Last Accessed

Papers 1, 2 and 3

N11, R3, R4, R5, R6, R7, R8 - Ratios, fractions and proportional sharing

Ratio notation and multiplicative relationships

Part-to-part and part-to-whole fractions

Sharing quantities and recovering totals

Equal proportions and linear relationships

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