GCSE · Maths · AQA · Spec 8300

Sharing a quantity in a given ratio

Amy paid £3 towards a raffle ticket and Ben paid £5. It wins £40. Half each isn't fair, so how do you split £40 in the ratio 3 : 5?

Ratio · The bar model

£40 shared between Amy and Ben
£0£5£10£15£20£25£30£35£40drag to give Amy more boxes →

Amy's boxes: 3/8. Amy gets £15. Ben gets £25. One box £5

Fraction3/8Amy : Ben3 : 5Amy gets£15Ben gets£25One box£5

The whole bar is £40, cut into 8 equal boxes. Drag the handle to choose how many boxes Amy gets, and Ben gets the rest. Try setting Amy : Ben to 3 : 5. What does each of them get?

Exam line: To share in the ratio a : b, add a + b to count the equal boxes. Divide the total by that number to get one box, then multiply by a and by b.

What would you do first?

Priya and Sam share £35 in the ratio 2 : 5

Priya and Sam share £35 in the ratio 2 : 5. Priya gets the 2 parts and Sam gets the 5.

Which first move is closest to what you would do?
How sure are you?

Exam line: Count ALL the equal parts first: 2 + 5 = 7. Every box in the bar has to be the same size, because that is what lets one division find them all.

Your turn · Given one part

When the number you're given isn't the total

Orange squash is mixed with water in the ratio 1 : 6. Ella uses 45 ml of squash. How much water does she need, and how much drink does she make?

  1. Put the ratio in a table: Squash 1, Water 6. The 45 ml goes under Squash, because it's the squash on its own, not the whole drink.This is the decision that matters. 45 ml is one person's share here, not the total.
  2. missing step
Which line is step 2?

Exam line: Before you divide, ask what the given amount is: the whole, or one person's share? If it's one share, scale the ratio up from it.

Spot the slip

Writing a split as a ratio

Leo has 350 g of flour. He uses 150 g for bread and the rest for a cake. Write the ratio of bread flour to cake flour in its simplest form.

A student's answer — which line goes wrong?

Exam line: Before you write a ratio, name what it compares. Part : part uses the two pieces. Part : whole uses one piece and the total.

Predict, then check

Same shampoo, two bottle sizes. Commit to an answer before you work anything out.

A 300 ml bottle costs £2.40. A 500 ml bottle of the same shampoo costs £4.25. Which is better value?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A ratio share is always made of equal boxes. Work out what one box is worth, and everything else is counting.

What you need to know

  • A ratio a : b splits the whole into a + b equal parts. One part = total ÷ (a + b). Then multiply one part by a and by b.
  • Part : part compares the two shares (Amy : Ben = 3 : 5). Part : whole compares one share with the total (Amy : whole = 3 : 8, so Amy gets 3/8).
  • If you're given one share instead of the total, it stands for that person's number of parts. Scale the ratio up from it, and a ratio table keeps the working tidy.
  • To write a split as a ratio, put the two amounts in the order the question asks, then simplify by dividing both by a common factor.
  • For best buy, compare like with like: scale both deals to the same amount, or find the price of one unit of each.

The big picture

To share an amount in a ratio, picture a bar made of equal boxes. Add the ratio numbers to count the boxes, divide the total by that to find what one box is worth, then multiply by each person's number of boxes. The same 'value one box' idea works when you're given one share instead of the total, when you write a split as a ratio, and when you decide which pack is better value.

Key points

1In a bar model every box must be the same size. Unequal boxes don't show the ratio correctly.
2The bigger ratio number always gets the bigger share.
3Change the total and every box changes with it: share £80 in 3 : 5 and each box is £10, so the shares double while the ratio stays 3 : 5.
4Ratios scale by multiplying or dividing both numbers, never by adding the same amount to both.
5Check any answer twice: the shares add back to the total, and they simplify back to the ratio.

Worked example

Problem

A garden centre has 840 plants. The ratio of roses to ALL the plants is 3 : 7. How many of the plants are not roses?

⚠ Watch out

Dividing the total by one ratio number. For £40 in 3 : 5, £40 ÷ 3 and £40 ÷ 5 are nobody's share, and they don't even add up to £40. The 3 and 5 count boxes: add them to get 8, then divide by 8.

🧠

Memory hook

Count the boxes, value one box, then fill them in.

✓

Check yourself

Share £72 in the ratio 5 : 3. What fraction of the £72 is the larger share? Check that your two shares add back to £72.

Flashcards

(10)
In the ratio 3 : 5, how many equal parts is the whole split into?
8. Add the ratio numbers: 3 + 5.
When you share an amount in a ratio, what do you work out first?
The value of one part: the total ÷ the sum of the ratio numbers.
What's the difference between a part : part ratio and a part : whole ratio?
Part : part compares the two shares with each other. Part : whole compares one share with the total.
If boys : girls = 2 : 3, what fraction of the group are girls?
3/5. The girls are 3 of the 2 + 3 = 5 parts.
Why must every box in a ratio bar model be the same size?
Otherwise the bar doesn't show the ratio correctly. One box has to stand for the same amount everywhere.
You're given one person's share, not the total. What do you do?
Work out how many parts that share is, find one part, then scale up to the other share or the total.
Write '18 red and 24 blue' as a ratio of red to blue in its simplest form.
18 : 24, then divide both by 6 to get 3 : 4.
What two checks tell you a ratio share is right?
The shares add up to the total, and they simplify back to the given ratio.
What are two ways to decide which pack is better value?
Scale both to the same amount and compare prices, or find the cost of one unit (such as 100 ml) of each.
Is the bigger pack always the better buy?
No. It only wins if the same amount costs less, so compare like with like.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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