C1.10; E1.10 - Ratio and sharing | Cambridge IGCSE International Mathematics

Two people share $90 in the ratio$ 2:3$. Does the first person get two thirds of the money? Ratio compares parts, so you must first decide what each number refers to. This lesson develops simplifying ratios, sharing quantities and connecting part-to-part and part-to-whole comparisons in C1.10 and E1.10.

Simplifying ratios

A ratio describes relative amounts in a stated order. If red paint to white paint is $2:3$ , every two equal measures of red go with three of white. Multiplying both amounts by the same positive factor keeps the mixture unchanged.

Equivalent ratios

Equivalent ratios multiply or divide every part by the same positive number. A ratio is in its simplest whole-number form when its parts are whole numbers with no common factor greater than 1.

Before simplifying measurements, use the same unit in every part. There are $1000\,\mathrm{g}$ in $1\,\mathrm{kg}$ , so $1\,\mathrm{kg}:500\,\mathrm{g}$ is $1000:500=2:1$ , rather than $1:500$ . For lengths, $1\,\mathrm{m}=100\,\mathrm{cm}$ : multiply metres by 100 to express them in centimetres.

Worked example 1: simplifying with different units

Question. Write the ratio $1.2\,\mathrm{kg}:450\,\mathrm{g}$ in its simplest form.

Plan. Convert both masses to grams, then divide by their highest common factor.

\begin{aligned} 1.2\,\mathrm{kg}:450\,\mathrm{g}&=1200\,\mathrm{g}:450\,\mathrm{g}\\ &=1200:450\\ &=8:3 &&\text{divide both parts by 150} \end{aligned}

Answer. $8:3$ .

Check. $8\times150=1200$ and $3\times150=450$ ; the same multiplier restores both masses.

Add a third quantity. The rule stays the same, but every division must now apply to all three parts.

Worked example 2: a three-part ratio

Question. Write $1.2\,\mathrm{kg}:450\,\mathrm{g}:750\,\mathrm{g}$ in its simplest form.

Plan. Use grams throughout, then divide all three parts by 150.

\begin{aligned} 1.2\,\mathrm{kg}:450\,\mathrm{g}:750\,\mathrm{g}&=1200:450:750 &&\text{all masses in grams}\\ &=8:3:5 &&\text{divide each part by 150} \end{aligned}

The parts 8, 3 and 5 have no common factor greater than 1, so the ratio is fully simplified.

Answer. $8:3:5$ .

Check. Multiplying all three parts by 150 gives the original masses in grams. All pairs retain their original relative size.

For fractional or decimal parts, first multiply every part by a convenient common number to obtain whole numbers, then simplify. For example, $\frac12:\frac34=2:3$ after multiplying both parts by 4.

Watch out

Do not subtract the same amount from every part. $8:3$ and $7:2$ are not equivalent: $8/3\ne7/2$ . Scaling requires multiplication or division by a common factor.

Convert the units before finding the common factor.

Sharing an amount

To share a total in the ratio $2:3$ , imagine five equal parts: two go to the first person and three to the second. Adding the ratio parts finds the number of equal shares in the whole amount.

Sharing in a ratio

For a total $T$ shared in the positive ratio $a:b$ ,

\text{one part}=\frac{T}{a+b}.

Multiply one part by $a$ and by $b$ for the two shares. For three parts, divide by $a+b+c$ instead. This method is not on the supplied formula list: learn it.

The equal parts are a measuring tool: they do not mean the two people receive equal amounts.

DIAGRAM: asset_name: Sharing ninety in the ratio two to three; asset_slug: ratio_ninety_bar; caption: Five equal parts worth $18 each give shares of$ 36 and $54.; description: An exact bar divided into five equal cells. The first two pale blue cells are bracketed as first share, two parts, thirty-six dollars. The last three pale purple cells are bracketed as second share, three parts, fifty-four dollars. Each cell is labelled eighteen dollars and the total ninety dollars appears above the bar.] ![Five equal parts worth$ 18 each give shares of $36 and$ 54.

Worked example 3: a two-part share

Question. Divide $90 between Ali and Beth in the ratio$ 2:3$.

Plan. Find the value of one of the five equal parts, then multiply.

\begin{aligned} \text{total parts}&=2+3=5\\ \text{one part}&=90\div5=18\\ \text{Ali's share}&=2\times18=36\\ \text{Beth's share}&=3\times18=54 \end{aligned}

Answer. Ali gets $36 and Beth gets$ 54.

Check. $36+54=90$ , and $36:54=2:3$ . Both the total and the ratio agree.

Keep the total fixed and add a third recipient. The parts are now smaller because the total is divided into nine equal pieces.

Worked example 4: a three-part share

Question. Divide $90 between Ali, Beth and Chen in the ratio$ 2:3:4$.

Plan. Add all three ratio parts before dividing the total.

\begin{aligned} \text{total parts}&=2+3+4=9\\ \text{one part}&=90\div9=10\\ \text{shares}&=2\times10,\quad3\times10,\quad4\times10 \end{aligned}

Answer. Ali gets $20, Beth gets$ 30 and Chen gets $40.

Check. $20+30+40=90$ , and dividing the three shares by 10 gives $2:3:4$ .

The opening question is settled: the first person receives $2/(2+3)=2/5$ of the money, rather than $2/3$ . The 3 compares the second person's share with the first; it is not the total.

Watch out

The denominator for a share of the whole is the sum of the parts. Dividing $90 by 3 in a$ 2:3$ share treats only the second person's parts as the whole.

Check the total as well as the relative sizes.

Connecting ratios

A ratio such as red to white $=2:3$ compares one part with another part. It also tells you that red is $2/5$ of the total and white is $3/5$ of the total. Read whether a given quantity is one component or the whole mixture before choosing a divisor.

Part-to-part and part-to-whole

In a two-part ratio $a:b$ , the part-to-part comparison is $a:b$ . The fractions of the whole are $\frac{a}{a+b}$ and $\frac{b}{a+b}$ .

When one component is known, use its own number of ratio parts to find the common multiplier. When the total is known, use the sum of the ratio parts.

Worked example 5: one component is known

Question. Red and white paint are mixed in the ratio $2:3$ . There are $450\,\mathrm{mL}$ of white paint. Find the red paint needed and the total volume.

Plan. The white paint represents three parts, so divide 450 by 3.

\begin{aligned} \text{one part}&=450\div3=150\,\mathrm{mL}\\ \text{red paint}&=2\times150=300\,\mathrm{mL}\\ \text{total}&=300+450=750\,\mathrm{mL} \end{aligned}

Answer. $300\,\mathrm{mL}$ of red paint; $750\,\mathrm{mL}$ in total.

Check. $300:450=2:3$ , and $300/750=2/5$ . The part-to-part and part-to-whole descriptions agree.

Keep the mixture ratio but change what is given: the next volume is the whole mixture. It corresponds to five parts, not three.

Worked example 6: the whole is known

Question. Red and white paint are mixed in the ratio $2:3$ . Find the red and white volumes in $900\,\mathrm{mL}$ of mixture.

Plan. Use the fractions $2/5$ and $3/5$ of the total.

\begin{aligned} \text{red paint}&=\frac25\times900=360\,\mathrm{mL}\\ \text{white paint}&=\frac35\times900=540\,\mathrm{mL} \end{aligned}

Answer. $360\,\mathrm{mL}$ red and $540\,\mathrm{mL}$ white.

Check. $360+540=900$ , and $360/540=2/3$ , so the required part-to-part ratio is preserved.

The same reasoning scales recipes: if 2 parts flour go with 3 parts oats, choosing the flour amount fixes the common multiplier for the oats. It will also support map-scale and proportional-reasoning problems later.

Watch out

In a $2:3$ mixture, red divided by white is $2/3$ , while red divided by total is $2/5$ . Label the quantities before choosing a fraction.

Decide which component the given count describes.

Quick Recap

Put measurements in common units before simplifying a ratio.
Scale every ratio part by the same multiplier.
To share a total, add the parts, find one part, then multiply.
Distinguish a known component from a known whole; a ratio $a:b$ gives a first-share fraction $a/(a+b)$ .

The opening $2:3$ share of $90 gives$ 36 and $54, with the first person receiving two fifths of the whole.

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C1.10; E1.10 - Ratio and sharing

Simplifying ratios

Sharing an amount

Connecting ratios

Quick Recap

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