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# Sharing in a ratio

Number Foundations · Ratios

<!-- Speaker note: Students share a total in a given ratio by cutting it into equal parts (add the parts, find one part, multiply), with two and three people, and work backwards from a share to the total (AC9M7N08, node 3). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** share $60 in the ratio 2 : 3. Add the parts (5), find one part ($60 ÷ 5 = $12), then multiply: $24 and $36.

- I can share an amount in a given ratio, with two or three people.
- I can find the total number of parts and the value of one part.
- I can work backwards from one share to the other share or the total.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need equivalent ratios (previous lesson) and division. The one idea is the three-step routine: add the parts, divide the total by the number of parts to get one part, then multiply by each person's number. Insist on the check that the shares add back to the total. The commonest mistake is dividing the total by each term (Dee, $30 and $20), which is caught by the check. Working backwards (Modify) finds one part from the known share first. -->

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## Why it matters

> Ana and Ben earn $60 washing cars. Ana did 2 cars, Ben did 3. They agree to share the money in that ratio. How much does each get?

<!-- Speaker note: To **share a total in a ratio**, think of the total cut into equal **parts**: 1. **Add the parts:** $2 + 3 = 5$ parts. 2. **Find one part:** $\$60 \div 5 = \$12$. 3. **Multiply:** Ana gets $2 \times \$12 = \$24$, Ben $3 \times \$12 = \$36$. Check: $\$24 + \$36 = \$60$. The shares must add to the total. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**$60 is shared between Ana and Ben in the ratio $2 : 3$. How much does Ben get?**

- $30
- $24
- $36
- $20

<!-- Speaker note: What happens: $36. There are 2 + 3 = 5 parts, each $12, and Ben has 3 parts. $30 is an equal split; $24 is Ana's share. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose $20 divided by Ben's 3; $30 split equally. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fsharing-in-a-ratio%2Fstep-2&code=widget%3A+ratio%0Amode%3A+share%0Alabels%3A+Ana%2C+Ben%0Aa%3A+2%0Ab%3A+3%0Aslider+total%3A+10%2C+200%2C+10%2C+10%0Aunits%3A+%24%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/sharing-in-a-ratio/?full=step-2#step-2)

<!-- Speaker note: Change the total and watch the parts stay equal. -->

<!-- Speaker note: Teacher: ask how many equal parts the bar is cut into before they touch the total. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* $36. There are 2 + 3 = 5 parts, each $12, and Ben has 3 parts. $30 is an equal split; $24 is Ana's share.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## One part

Here is the whole routine for $60 in the ratio 2 : 3.

Now try one: $40 in the ratio 1 : 3.

$$2 + 3 = 5$$

1) $\$60 \div 5 = \$12$ — One part is the total divided by the number of parts.
2) $\text{Ana: } 2 \times \$12 = \$24$ — Each share is its number of parts times one part.
3) $\text{Ben: } 3 \times \$12 = \$36$ — Ben has 3 parts.
4) $\$24 + \$36 = \$60$ — Check that the shares add back to the total.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 2 + 3 = 5 and say "Add the parts of the ratio." Line 1: write \$60 ÷ 5 = \$12 and say "One part is the total divided by the number of parts." Line 2: write Ana: 2 × \$12 = \$24 and say "Each share is its number of parts times one part." Line 3: write Ben: 3 × \$12 = \$36 and say "Ben has 3 parts." Line 4: write \$24 + \$36 = \$60 and say "Check that the shares add back to the total." -->

<!-- Speaker note: Teacher: have students say why Dee's shares do not add to $60. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## One part

**$40 is shared in the ratio $1 : 3$. How much is one part, in dollars?**

* Answer: 10

<!-- Speaker note: 1 + 3 = 4 parts. $40 ÷ 4 = $10 for one part. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Dee shares $60 between Ana and Ben in the ratio 2 : 3. Click the line where her working goes wrong.**

```
 1  Share $60 between Ana and Ben in the ratio 2 : 3.
 2  Ana: 60 ÷ 2 = $30
 3  Ben: 60 ÷ 3 = $20
 4  Check: 30 + 20 = $50, near enough to 60.
```

* Answer: line 2

<!-- Speaker note: Line 2: dividing $60 by Ana's 2 does not give her share. First add the parts (2 + 3 = 5), find one part ($12), then Ana has 2 × $12 = $24. The check on line 4 shows the shares do not add to $60. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The bigger share

**35 stickers are shared in the ratio $3 : 4$. How many stickers are in the bigger share?**

* Answer: 20

<!-- Speaker note: 3 + 4 = 7 parts. 35 ÷ 7 = 5 stickers a part. The bigger share is 4 × 5 = 20. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Three people, and working backwards



- **Three shares:** 
- **Work backwards:** 
- **The whole amount:** 
- **A mortar mix:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/sharing-in-a-ratio/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Three people share the same way; working backwards from one share finds the total by finding one part first. For students who need more support: Keep the bar on screen and read each bracket aloud: 'Ana has two parts.' For early finishers: A profit is shared among three partners in the ratio 2 : 3 : 5. The biggest partner gets $1200 more than the smallest. How much was the profit? The hard part, on purpose: Starting by adding the parts, not dividing by one of the numbers. Say 'how many equal parts is the total cut into?' before any dividing. When students get stuck: A student divides the total by each term (Dee's slip). Ask: Do the two shares add to the total? Fix: Add the parts first: the total is cut into 5 parts, not 2 and 3 parts separately. / A student gives one part as the answer. Ask: How many parts does this person have? Fix: Multiply one part by the person's number of parts. / A student shares $60 in 2 : 3 and gets $30 and $30. Ask: How many parts does each person have? Fix: Ana has 2 parts, Ben has 3. Draw the bar. -->

<!-- Speaker note: Teacher: for working backwards, ask 'what is one part?' before 'what does Noor get?' -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The bigger share

**$72 is shared in the ratio $5 : 4$. What is the bigger share, in dollars?**

* Answer: 40

<!-- Speaker note: 5 + 4 = 9 parts. 72 ÷ 9 = 8. The bigger share is 5 × 8 = 40. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The smaller share

**45 marbles are shared in the ratio $2 : 3$. How many are in the smaller share?**

* Answer: 18

<!-- Speaker note: 2 + 3 = 5 parts. 45 ÷ 5 = 9. The smaller share is 2 × 9 = 18. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Three people

**$120 is shared among three people in the ratio $1 : 2 : 3$. What is the largest share, in dollars?**

* Answer: 60

<!-- Speaker note: 1 + 2 + 3 = 6 parts. 120 ÷ 6 = 20. The largest has 3 parts: 60. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: simplest form

**Review: simplify $18 : 24$. It is $3 : ?$**

* Answer: 4

<!-- Speaker note: Divide both by 6: 3 : 4. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: exactly $15

Then find two different ratios (with their totals) that give the same share.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/sharing-in-a-ratio/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing a ratio and a total so that one share is exactly $15 shows the share is parts times one part, and many choices work. Hints, in order: Ana's share is a × one part. / Try 1 : 1 with a total of $30. -->

<!-- Speaker note: Teacher: collect ratio and total pairs that give $15. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Share $84 between Sam and Tia in the ratio 3 : 4. How much does Tia get?**

* 3 + 4 = 7 parts. 84 ÷ 7 = 12. Tia has 4 parts: $48.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: Modelling with money and with measurements, using ratios with percentages and fractions. Where it leads: unit rates, scale drawings, similar shapes, proportion and direct variation in Years 8 and 9. -->