Sharing in a given ratio

The idea

To divide a quantity x in the ratio m : n, split it into m + n equal groups of size x/(m + n); the parts are m × x/(m + n) and n × x/(m + n).

Sharing, but not equally

12 counters are to be shared between two people, but not equally. For every 3 counters the first person takes, the second person takes 1.

How many counters does each person end up with?

• 9 and 3
• 8 and 4
• 6 and 6

Deal them out

Each press of Deal a round hands the first person m counters and the second person n. Every round is one column. Deal until the pile is empty, in all three piles, then look at the rows.

Equal groups, then m of them and n of them

To divide a quantity x in the ratio m : n, split it into m + n equal groups of size x/(m + n); the parts are m × x/(m + n) and n × x/(m + n).

Each round hands out m + n counters, so the pile is empty after x ÷ (m + n) rounds. That number is also the size of every group, the length of each row.

Share 42 in 4 : 3. There are 4 + 3 = 7 groups of 42 ÷ 7 = 6, so the parts are 4 × 6 = 24 and 3 × 6 = 18. Check: 24 : 18 :: 4 : 3.

Prashanti put in ₹75,000 and Bhuvan ₹25,000, a ratio of 3 : 1. They share a ₹4,000 profit in that ratio: 4 groups of ₹1,000, so ₹3,000 and ₹1,000.

Notes

To divide a quantity x in the ratio m : n, split it into m + n equal groups of size x/(m + n); the parts are m × x/(m + n) and n × x/(m + n).

Check yourself

54 sweets are shared between Amit and Beena in the ratio 4 : 5. How many does Beena get?

Answer: 30

4 + 5 = 9 groups, and 54 ÷ 9 = 6 in each group. Beena gets 5 × 6 = 30 sweets (Amit gets 4 × 6 = 24).

Two friends invest ₹60,000 and ₹40,000 in a stall. The profit is ₹5,000, shared in the ratio of their investments. How much does the friend who invested ₹40,000 get? (in ₹)

Answer: 2000

60000 : 40000 = 3 : 2. There are 3 + 2 = 5 groups of 5000 ÷ 5 = ₹1,000. The ₹40,000 friend gets 2 × 1000 = ₹2,000.

A 40 kg mixture has sand and cement in the ratio 3 : 1. How many kg of cement must be added to make the ratio of sand to cement 5 : 2?

Answer: 2 kg

3 + 1 = 4 groups of 10 kg, so sand 30 kg and cement 10 kg. For 5 : 2 :: 30 : ?, the cement is 2/5 of 30 = 12 kg. Add 12 − 10 = 2 kg.

48 is to be shared in the ratio 5 : 3. Which calculation gives the size of one group?

• 48 ÷ 8 — correct. Yes! There are 5 + 3 = 8 groups, so each has 48 ÷ 8 = 6.
• 48 ÷ 5. That divides by one part only. Divide by the total number of groups, 5 + 3.
• 48 ÷ 3. That divides by one part only. Divide by the total number of groups, 5 + 3.
• 48 ÷ 2. That is an equal share between two people. The ratio is 5 : 3, so there are 8 groups.