Sharing in a ratio

Share $60 in the ratio 2 : 3. Add the parts (5), find one part ($60 ÷ 5 = $12), then multiply: $24 and $36.

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?

To share a total in a ratio, think of the total cut into equal parts:

Add the parts: $2 + 3 = 5$ parts.
Find one part: $$60 \div 5 = $12$.
Multiply: Ana gets $2 \times $12 = $24$, Ben $3 \times $12 = $36$.

Check: $$24 + $36 = $60$. The shares must add to the total.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

How much does Ben get?

Should Ben get more or less than half of the money?

$60 is shared between Ana and Ben in the ratio

2 : 3

. How much does Ben get?$30$24$36$20

TryTry it on the board and compare the result with your prediction.

Cut it into parts

Change the total and watch the parts stay equal.

Try it yourselfYour edits are saved in this browser.

Set the total to $60. How much is one part, and how much does Ben get?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Parts first

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

Worked exampleBefore each step, say what you think comes next.

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

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

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

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

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

ModifyChange the problem on purpose and see what happens.

Three people, and working backwards

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Three friends share $90 in the ratio 1 : 2 : 3. How much does the third friend get?

Use the board. This ticks when you get there.

Mia and Noor share money in the ratio $3 : 5$ . Mia gets $24. How much does Noor get, in dollars?

How much money did Mia and Noor share altogether, in dollars?

Mortar is made from cement and sand in the ratio $2 : 5$ . How many kilograms of cement are in 14 kg of mix?

PractiseA few on your own, to check you have it.

Sharing in a ratio

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

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

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

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Exactly $15

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

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Need a hint?

Ana's share is a × one part.

Another hint

Try 1 : 1 with a total of $30.

Choose a ratio and a total so that Ana's share is exactly $15.

Use the board. This ticks when you get there.

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

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.

Running the lesson

Predict: students who chose $20 divided by Ben's 3; $30 split equally.
Run: ask how many equal parts the bar is cut into before they touch the total.
Investigate: have students say why Dee's shares do not add to $60.
Modify: for working backwards, ask 'what is one part?' before 'what does Noor get?'
Make: collect ratio and total pairs that give $15.

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.

Stretch and support

Stretch: 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?

Support: Keep the bar on screen and read each bracket aloud: 'Ana has two parts.'

Where it 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.

What each step is for

how much does Ben get?: Committing shows whether they split equally ($30), divide by Ben's number ($20) or divide by all the parts ($36).
cut it into parts: Seeing the bar cut into five equal parts of $12, with brackets for each person, shows the shares as parts of the total.
parts first: Find the total parts first, then one part; Dee divides the total by each person's number instead. Shares add back to the total.
three people, and working backwards: Three people share the same way; working backwards from one share finds the total by finding one part first.
sharing in a ratio: Two-part shares with both a bigger and a smaller share, a three-part share, and a review of simplest form.
exactly $15: 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.

Now you can

Share an amount in a given ratio, with two or three people.
Find the total number of parts and the value of one part.
Work backwards from one share to the other share or the total.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson23. Modelling with moneyDiscounts, markups and profit: formulate the problem, choose fractions, percentages or ratios, solve, and say what the answer means in dollars.

Source: content/maths/number/sharing-in-a-ratio/lesson.md