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MST-12-S2-05 ~35 min ⚡ +90 XP available

Ratios, Simplest Form and Dividing a Quantity

A ratio compares two quantities measured in the same unit. Simplify it, read it as a fraction, and split any total in a given ratio.

Today's hook, Two friends put in $\$50$ and $\$30$ to buy a lottery ticket. It wins $\$480$. Splitting it evenly feels unfair. How much should each get?

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Orient and prepare

Capture your first idea and preview the language of ratios.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any question in this focus area.

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Think first, your gut answer

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Two friends put in $\$50$ and $\$30$ to buy a lottery ticket. It wins $\$480$. Splitting it evenly feels unfair to the one who paid more.

Without calculating write down roughly how you would split it, and say what you are comparing.

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What you will be able to do

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Write a ratio and say how it relates to a fraction
Express a ratio in simplest form
Find the ratio of two quantities, converting to the same unit first
Divide a quantity in a given ratio, for two or three parts
Use ratios in practical problems such as recipes, mixes and shared costs

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Preview the vocabulary

Review the essential terms before using them in the worked methods.

Key terms

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Ratio: A comparison of two or more quantities measured in the same unit, written with a colon. Like this: a mix of 2 buckets of cement to 3 buckets of sand is written $2:3$.
Simplest form: A ratio where the numbers share no common factor except 1, so it cannot be reduced further. Like this: $18:24$ has a common factor of 6, and dividing both by 6 gives $3:4$, which is simplest form.
Part: One share of the total when a quantity is split. Add the numbers in the ratio to get the number of parts. Like this: in $5:3$ there are $5+3=8$ parts, so each part is one eighth of the total.
Equivalent ratio: A ratio giving the same comparison, made by multiplying or dividing every term by the same number. Like this: $3:4$, $6:8$ and $30:40$ are all equivalent, and only $3:4$ is in simplest form.

Connect ratios and fractions

Simplify ratios and connect each comparison to a fraction.

A ratio is a fraction wearing different clothes

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A ratio compares parts with each other. A fraction compares one part with the whole. Both describe the same split, and moving between them is what most exam questions are really testing.

In the ratio $3:4$ there are $3+4=7$ parts altogether. The first quantity is $\tfrac{3}{7}$ of the whole and the second is $\tfrac{4}{7}$. Notice the ratio numbers become the numerators, and the total number of parts becomes the denominator.

$a:b \;\Rightarrow\; \text{first} = \dfrac{a}{a+b} \text{ of the whole}$

Worked example 1, simplest form

Simplify $18:24$

Find the highest common factor of 18 and 24.

$\text{HCF}(18,24)=6$

6 divides both exactly, and nothing larger does.

$18 \div 6 = 3$, $\;24 \div 6 = 4$

Divide every term by the HCF.

$18:24 = 3:4$

Check by multiplying back: $3 \times 6 = 18$ and $4 \times 6 = 24$.

Same unit first, always. A ratio only makes sense when both quantities are in the same unit. Convert before you simplify, never after.

Worked example 2, ratio of two quantities

Write $250\text{ g}$ to $1\text{ kg}$ as a ratio

The units differ, so convert first.

$1\text{ kg} = 1000\text{ g}$

Work in grams, the smaller unit, to avoid decimals.

$250:1000$, $\;\text{HCF}=250$

Both terms divide exactly by 250.

$250:1000 = 1:4$

Check: $1 \times 250 = 250$ and $4 \times 250 = 1000$.

In your book. Write $18:24 = 3:4$ and beside it $\tfrac{3}{7}$ and $\tfrac{4}{7}$. Say in one sentence why the denominator is 7 and not 4.

Did you get this? True or false: $250\text{ g}:1\text{ kg}$ simplifies to $250:1$.

Divide a quantity in a ratio

Find the total number of parts, then allocate each share.

Dividing a quantity in a given ratio

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This is the hook question, and it is three steps every time: count the parts, find one part, then multiply out.

Worked example 3, the lottery ticket

Share $\$480$ in the ratio $5:3$

The friends paid $\$50$ and $\$30$, which is $5:3$ in simplest form.

Parts $= 5 + 3 = 8$

Step 1, add the terms to get the number of equal parts.

One part $= 480 \div 8 = 60$

Step 2, divide the total by the number of parts.

$5 \times 60 = \$300$, $\;3 \times 60 = \$180$

Step 3, multiply out. Check: $300 + 180 = 480$, the whole prize.

Always check the total. The shares must add back to the original quantity. It catches almost every arithmetic slip in this topic.

Worked example 4, three parts

Share $63\text{ kg}$ in the ratio $2:3:4$

The method does not change when there are three terms.

Parts $= 2 + 3 + 4 = 9$

Add all three terms.

One part $= 63 \div 9 = 7\text{ kg}$

Divide the total by 9.

$14\text{ kg}$, $\;21\text{ kg}$, $\;28\text{ kg}$

Check: $14 + 21 + 28 = 63$, the whole mass.

In your book. Copy the three steps as a numbered list: count the parts, find one part, multiply out. Add the checking line underneath.

Quick check. A quantity is shared in the ratio $4:5$. How many equal parts is it split into?

Apply ratios in context

Transfer the method to practical ratio problems and check your reasoning.

Ratios in practice

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Exam questions dress ratios up as recipes, concrete mixes, paint colours and shared costs. The wording changes; the three steps do not.

Watch for the question that gives you one part rather than the total. If a mix is cement to sand $1:3$ and you are told there is $12\text{ kg}$ of sand, then $12\text{ kg}$ is three parts, so one part is $12 \div 3 = 4\text{ kg}$ and you need $4\text{ kg}$ of cement.

Order matters

Cement to sand $1:3$ is not the same as sand to cement $1:3$. Write the labels above the numbers.

Total or part?

Decide whether the number you are given is the whole quantity or just one share, before dividing anything.

Units on the answer

Shares carry the unit of the original quantity. Marks are lost for bare numbers.

In your book. Write the cement and sand example out in full, labelling which number is one part and which is the total.

Fill the gaps. To share $\$720$ in the ratio $3:5$: parts $=$ , one part $= \$$, shares $= \$$ and $\$$.

Show what you can do

Answer the exam-style questions, then compare your working with the model answers.

Quick-check drill

work before revealing answers

Choose an option for each fixed drill question, then reveal the concept-labelled explanations below.

Drill 1. Express $45:60$ in simplest form.

$3:4$
$9:12$
$15:20$
$5:4$

Drill 2. Write $400\text{ mL}$ to $2\text{ L}$ as a ratio in simplest form.

$2:5$
$1:5$
$1:4$
$5:1$

Drill 3. $\$360$ is divided in the ratio $4:5$. What is the larger share?

$\$160$
$\$180$
$\$200$
$\$225$

Drill 4. A quantity is divided in the ratio $2:3$. What fraction of the whole is the first share?

$\tfrac{2}{3}$
$\tfrac{2}{5}$
$\tfrac{3}{5}$
$\tfrac{1}{2}$

Drill 5. A concrete mix has cement to sand in the ratio $1:3$. A batch uses $12\text{ kg}$ of sand. How much cement is needed?

$3\text{ kg}$
$4\text{ kg}$
$9\text{ kg}$
$36\text{ kg}$

Short answer

ApplyBand 43 marks

SA 1. A biscuit recipe uses flour, sugar and butter in the ratio $6:2:1$ and makes a total mass of $450\text{ g}$. Find the mass of each ingredient. (3 marks)

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ApplyBand 43 marks

SA 2. One piece of timber is $84\text{ cm}$ long and another is $1.4\text{ m}$ long. (a) Write the ratio of the shorter to the longer in simplest form. (2 marks) (b) What fraction of the longer piece is the shorter piece? (1 mark)

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AnalyseBand 54 marks

SA 3. Three people share $\$1260$ in the ratio $2:3:4$. (a) Find each person's share. (3 marks) (b) How much more does the person with the largest share receive than the person with the smallest? (1 mark)

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📖 Comprehensive answers (click to reveal)

Simplifying a ratio. $\text{HCF}(45,60)=15$, so $45:60 = 3:4$. Options B and C are equivalent ratios but not in simplest form.

Converting units before comparing. $2\text{ L} = 2000\text{ mL}$, so $400:2000$, and $\text{HCF}=400$ gives $1:5$. Option D reverses the order.

Dividing in the ratio 4:5. Parts $=4+5=9$; one part $= 360 \div 9 = 40$; larger share $= 5 \times 40 = \$200$. Option A is the smaller share.

A ratio share as a fraction. Total parts $=2+3=5$, so the first share is $\tfrac{2}{5}$ of the whole. Option A confuses the ratio with a fraction.

Scaling a concrete mix. Sand is 3 parts, so one part $= 12 \div 3 = 4\text{ kg}$, and cement is 1 part $= 4\text{ kg}$. Option D multiplies instead of dividing.

SA 1 (3 marks): Parts $= 6+2+1 = 9$ [1]. One part $= 450 \div 9 = 50\text{ g}$ [1]. Flour $= 6 \times 50 = 300\text{ g}$, sugar $= 2 \times 50 = 100\text{ g}$, butter $= 1 \times 50 = 50\text{ g}$ [1]. Check: $300+100+50 = 450\text{ g}$.

SA 2 (3 marks): (a) $1.4\text{ m} = 140\text{ cm}$ [1]. $84:140$, and $\text{HCF}(84,140)=28$, so the ratio is $3:5$ [1]. (b) The shorter is $\tfrac{84}{140} = \tfrac{3}{5}$ of the longer [1].

SA 3 (4 marks): (a) Parts $= 2+3+4 = 9$ [1]. One part $= 1260 \div 9 = 140$ [1]. Shares are $\$280$, $\$420$ and $\$560$ [1]. Check: $280+420+560 = 1260$. (b) $560 - 280 = \$280$ more [1].

Retrieve, reflect and finish

Revisit your opening idea, then use the topic challenge and mark the lesson complete.

Revisit your thinking

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Go back to what you wrote in section 01. The friends paid $\$50$ and $\$30$, a ratio of $5:3$, and the fair split of $\$480$ is $\$300$ and $\$180$.

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Focus area overview · Year 12 · Maths Standard