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Sharing in a ratio

Ratio, rates and money · level 3, lesson 3 of 6

A question to start

Sam and Ria run a stall selling lemonade. Sam works for 3 hours and Ria works for 4 hours. At the end of the day they have made 35 dollars.

Sam says: "Let's split it half and half. That's 17.50 dollars each."

Ria says: "That's not fair. I worked longer."

Ria has a point. But how much should each of them get, so that the money matches the work? If you share 35 dollars so that for every 3 dollars Sam gets, Ria gets 4, every hour of work earns the same. How do you do that without guessing?

The idea

Sharing in rounds

Imagine the 35 dollars as a pile of loonies. Share them out in rounds. In each round, Sam takes 3 loonies and Ria takes 4. That uses \(3 + 4 = 7\) loonies per round.

How many rounds until the pile runs out? \(35 \div 7 = 5\) rounds.

  • Sam took 3 loonies in each of 5 rounds: \(3 \times 5 = 15\) dollars.
  • Ria took 4 loonies in each of 5 rounds: \(4 \times 5 = 20\) dollars.

Check: \(15 + 20 = 35\). ✓ And \(15 : 20\) divides by 5 to give \(3 : 4\). ✓

Each of them earned \(15 \div 3 = 5\) dollars per hour and \(20 \div 4 = 5\) dollars per hour. Same pay for each hour: that is why it is fair.

This "rounds" idea is the whole method. The rest of this lesson just gives it a quicker look.

Bar models

A bar model is a drawing that shows the ratio as boxes of equal size. For a \(3 : 4\) share, draw 3 boxes for Sam and 4 boxes for Ria, all the same size.

Sam 5 5 5 = 15 Ria 5 5 5 5 = 20 7 boxes = 35 one box = 35 ÷ 7 = 5

All 7 boxes together are worth 35 dollars. Since the boxes are equal, one box is \(35 \div 7 = 5\) dollars. Then Sam's 3 boxes are 15 dollars and Ria's 4 boxes are 20 dollars. One box in the bar model is one round in the loonie picture.

The three steps

To share an amount in the ratio \(a : b\):

  1. Add the parts to find how many boxes there are: \(a + b\).
  2. Find one part: divide the amount by the number of boxes.
  3. Multiply: one share is \(a\) boxes, the other is \(b\) boxes.

Then check: the shares add up to the amount, and they simplify to the ratio.

Step 2 is the key step. Almost every sharing problem is solved by finding the value of one part first.

What if you know something else?

The bar model is useful because it works even when you are not given the total. You always find one box first, using whatever you are told.

You know one share. Ali and Bea share stickers \(3 : 5\), and Ali gets 12. Ali's 3 boxes are 12 stickers, so one box is \(12 \div 3 = 4\). Bea's 5 boxes are \(5 \times 4 = 20\) stickers.

You know the difference. Ali and Bea share stickers \(3 : 5\), and Bea gets 10 more than Ali. In the bar model, Bea has 2 more boxes than Ali. Those 2 boxes are the 10 extra stickers, so one box is \(10 \div 2 = 5\). Ali gets \(3 \times 5 = 15\) and Bea gets \(5 \times 5 = 25\). Check: \(25 - 15 = 10\). ✓

Ali Bea 2 boxes = 10 more

Three people. The same steps work for \(a : b : c\). Add all three parts to count the boxes.

Why the answer is always fair

Sharing \(3 : 4\) means "Ria gets \(\tfrac43\) of what Sam gets", and the bar model makes that true by building both shares from the same-size box. You could also say Sam gets \(\tfrac37\) of the total and Ria gets \(\tfrac47\), since there are 7 boxes. Indeed \(\tfrac37 \times 35 = 15\) and \(\tfrac47 \times 35 = 20\). This is the part-to-whole idea from lesson 1 in action.

When it does not divide evenly

With money, the one-part value may come out in cents. Share 50 dollars in the ratio \(1 : 3\): there are 4 boxes, and one box is \(50 \div 4 = 12.50\) dollars. The shares are 12.50 and 37.50 dollars. That is fine.

But with things you cannot cut, like marbles or people, a share that is not a whole number tells you the ratio cannot be met exactly. Sharing 20 marbles in the ratio \(2 : 1\) gives one box \(= 20 \div 3 = 6\tfrac23\) marbles. You cannot do it exactly; the nearest is 13 and 7, or 14 and 6.

Worked examples

1. Share 56 dollars in the ratio \(3 : 4\).

  • Boxes: \(3 + 4 = 7\).
  • One box: \(56 \div 7 = 8\) dollars.
  • Shares: \(3 \times 8 = 24\) dollars and \(4 \times 8 = 32\) dollars.
  • Check: \(24 + 32 = 56\) ✓, and \(24 : 32 = 3 : 4\) (divide by 8) ✓.

2. Three people. Grandma gives 90 dollars to three grandchildren in the ratio of their ages, 4 : 5 : 6.

  • Boxes: \(4 + 5 + 6 = 15\).
  • One box: \(90 \div 15 = 6\) dollars.
  • Shares: \(4 \times 6 = 24\), \(5 \times 6 = 30\), \(6 \times 6 = 36\) dollars.
  • Check: \(24 + 30 + 36 = 90\) ✓.

3. Knowing one share. Flour and sugar are in the ratio \(5 : 2\). A baker uses 600 g of sugar. How much flour, and how much of the two together?

  • Sugar is 2 boxes, so one box is \(600 \div 2 = 300\) g.
  • Flour is 5 boxes: \(5 \times 300 = 1500\) g.
  • Together: 7 boxes, \(7 \times 300 = 2100\) g.

4. Knowing the difference. A bag has red and blue marbles in the ratio \(2 : 7\). There are 35 more blue marbles than red. How many marbles altogether?

  • Blue has \(7 - 2 = 5\) more boxes than red, and those are 35 marbles.
  • One box: \(35 \div 5 = 7\) marbles.
  • Altogether: \(2 + 7 = 9\) boxes, \(9 \times 7 = 63\) marbles (14 red, 49 blue). Check: \(49 - 14 = 35\) ✓.

5. Back to the lemonade stall. Sam (3 hours) and Ria (4 hours) earn 35 dollars. The fair split is \(3 : 4\): 15 dollars for Sam and 20 dollars for Ria. Half-and-half would have given Sam 17.50 dollars, which is \(17.50 \div 3 \approx 5.83\) dollars per hour, while Ria would get only \(17.50 \div 4 = 4.375\) dollars per hour, about 4.38. The ratio split gives both of them 5 dollars per hour.

Try it

  1. Share 40 dollars in the ratio \(3 : 5\).
  2. Share 72 stickers in the ratio \(1 : 2\).
  3. Share 100 dollars in the ratio \(2 : 3 : 5\).
  4. A school has teachers and pupils in the ratio \(1 : 18\). There are 25 teachers. How many pupils?
  5. Juice and water are mixed \(2 : 3\) to make 1.5 litres of drink. How much juice is used?
  6. Jo and Kai share money in the ratio \(4 : 7\). Kai gets 21 dollars more than Jo. How much does each get?
  7. Can 25 children be split into two teams in the ratio \(2 : 3\)? Can 26?
  8. A bead pattern repeats 2 orange, 1 green, 3 blue. A necklace uses 48 beads with no part pattern left over. How many of each colour?
Answers
  1. Boxes: 8. One box: \(40 \div 8 = 5\). Shares: 15 and 25 dollars. Check: \(15 + 25 = 40\).
  2. Boxes: 3. One box: \(72 \div 3 = 24\). Shares: 24 and 48 stickers.
  3. Boxes: \(2 + 3 + 5 = 10\). One box: 10 dollars. Shares: 20, 30 and 50 dollars.
  4. Teachers are 1 box, so one box is 25. Pupils: \(18 \times 25 =\) 450.
  5. 1.5 litres is 1500 mL. Boxes: 5, so one box is \(1500 \div 5 = 300\) mL. Juice is 2 boxes: 600 mL.
  6. Kai has \(7 - 4 = 3\) more boxes, worth 21 dollars, so one box is 7 dollars. Jo: \(4 \times 7 =\) 28 dollars; Kai: \(7 \times 7 =\) 49 dollars. Check: \(49 - 28 = 21\).
  7. Boxes: 5. For 25 children one box is 5: teams of 10 and 15, yes. For 26 children one box would be \(26 \div 5 = 5.2\) children, which is not a whole number, so no.
  8. One repeat has \(2 + 1 + 3 = 6\) beads, and \(48 \div 6 = 8\) repeats. Orange \(2 \times 8 = 16\), green \(1 \times 8 = 8\), blue \(3 \times 8 = 24\). Check: \(16 + 8 + 24 = 48\).

Common mistake

Dividing by one part instead of the total. To share 40 in the ratio \(3 : 5\), some people work out \(40 \div 3\) or \(40 \div 5\). But the 40 is spread over all \(3 + 5 = 8\) boxes, so one box is \(40 \div 8\). Draw the bar model and you will not make this mistake.

Forgetting to check. The two shares must add back to the total. If they do not, something went wrong. It takes five seconds and catches most errors.

Where it leads