Sharing in a given ratio

Say you and a partner must share 12 counters in the ratio 3 : 1. You could hand them out 3 and 1 at a time: three rounds, and your partner ends with 9 while you have 3.

That works, but imagine sharing 42 counters in the ratio 4 : 3 the same way. There's a faster route.

Think in groups. Your partner is to get 4 groups and you are to get 3, so there are 7 groups in all. Each group is 42 ÷ 7 = 6.

So your partner gets 4 × 6 = 24 and you get 3 × 6 = 18. Check: 24 + 18 = 42, and 24 : 18 reduces to 4 : 3.

The general rule: to divide a quantity x in the ratio m : n, there are m + n groups altogether, each of size x/(m + n). So the first part is m × x/(m + n) and the second is n × x/(m + n).

Try it on a business. Prashanti invested ₹75,000 and Bhuvan ₹25,000 in a food cart, and after a month they made ₹4,000 profit, to be shared in the ratio of their investments.

That ratio is 75000 : 25000, which reduces to 3 : 1. Total groups: 3 + 1 = 4. Group size: 4000 ÷ 4 = 1000. So Prashanti gets 3 × 1000 = ₹3,000 and Bhuvan gets ₹1,000.

A slightly harder one. A 40 kg mixture has sand and cement in the ratio 3 : 1. The sand is (3/4) × 40 = 30 kg and the cement (1/4) × 40 = 10 kg. How much cement must be added to make the ratio 5 : 2? Adding cement leaves the sand unchanged at 30 kg, so we need 5 : 2 :: 30 : ?, giving (2/5) × 30 = 12 kg of cement. There are already 10 kg, so add 2 kg.

Paint works the same way. Blue and yellow mixed 3 : 5 make green: 40 mL of green needs 15 mL blue and 25 mL yellow. Add 20 mL more yellow for a lighter shade and the blue stays at 15 mL, so the ratio becomes 15 : 45 = 1 : 3. Whatever you don't add to stays fixed.