7.6 Sharing, but Not Equally!

Suppose two friends share 20 marbles. Split equally and each gets 10, a ratio of 10 : 10 = 1 : 1. But what if they wish to share in the ratio 3 : 1?

Think of the ratio 3 : 1 as describing rounds: one friend takes 3 marbles at a time, the other takes 1, over and over, until the heap is gone. Each “round” hands out 3 + 1 = 4 marbles, and 20 = 5 \times 4, so there are 5 rounds: 3 : 1 \;\xrightarrow{\;\times 5\;}\; 15 : 5. One friend gets 15, the other 5, and 15 + 5 = 20.

For larger numbers, counting out rounds is slow. Here is the shortcut.

Dividing in a Ratio To divide a quantity x in the ratio m : n:

  1. Think of x as made of m + n equal groups (the total number of “parts”).
  2. The size of each group is \dfrac{x}{m+n}.
  3. The first share is m \times \dfrac{x}{m+n} and the second share is n \times \dfrac{x}{m+n}.

In symbols, \;x split as \;m \times \dfrac{x}{m+n} \;:\; n \times \dfrac{x}{m+n}\; is in the ratio m : n.

For example, sharing 56 marbles in the ratio 5 : 3: there are 5 + 3 = 8 groups, each of size 56 \div 8 = 7. So the shares are 5 \times 7 = 35 and 3 \times 7 = 21.

Worked Example 11: Splitting Profit

Anu invests ₹80{,}000, Vikram invests ₹20{,}000, and together they earn ₹6{,}000 profit, to be shared in the ratio of their investment. Find each share.

Investment ratio: 80000 : 20000 = 4 : 1. Number of parts = 4 + 1 = 5, and each part is worth $6000 = $ ₹1200. So \text{Anu} = 4 \times 1200 = \text{₹}4800, \qquad \text{Vikram} = 1 \times 1200 = \text{₹}1200.

Worked Example 12: Adjusting a Mixture

A 60 kg dry mix has flour and sugar in the ratio 4 : 1. How much sugar must be added to make the ratio 3 : 1?

First find the present amounts. With 4 + 1 = 5 parts in 60 kg, each part is 12 kg: \text{flour} = 4 \times 12 = 48 \text{ kg}, \qquad \text{sugar} = 1 \times 12 = 12 \text{ kg}. The flour stays 48 kg. We want the new ratio flour : sugar = 3 : 1, so 3 : 1 :: 48 : ? The first term scaled by 48 \div 3 = 16, so the sugar should be 1 \times 16 = 16 kg. There are already 12 kg, so add 16 - 12 = 4 kg of sugar.

Figure it Out: Sharing in a Ratio

Practice

  1. Divide ₹5{,}600 into two parts in the ratio 3 : 4.
  2. Acid and water are mixed in the ratio 1 : 4. A 300 mL bottle of this solution holds how much acid and how much water?
  3. Blue and yellow paints are mixed in the ratio 2 : 5 to make green. For 35 mL of green, how much of each colour is needed? If 15 mL of yellow is then added, what is the new blue-to-yellow ratio?
  4. Crisp dosas need rice and urad dal in the ratio 3 : 1. For 8 cups of mixture, how many cups of each?
  5. A tub of orange paint is red and yellow in the ratio 2 : 7. Another whole tub of yellow is added. Can you give the new red-to-yellow ratio exactly?
  1. Parts = 3 + 4 = 7; each part = 5600 \div 7 = 800. Shares: 3 \times 800 = \mathbf{₹2400} and 4 \times 800 = \mathbf{₹3200}.
  2. Parts = 1 + 4 = 5; each part = 300 \div 5 = 60 mL. Acid = 1 \times 60 = \mathbf{60} mL, water = 4 \times 60 = \mathbf{240} mL.
  3. Parts = 2 + 5 = 7; each part = 35 \div 7 = 5 mL. Blue = 2 \times 5 = \mathbf{10} mL, yellow = 5 \times 5 = \mathbf{25} mL. After adding 15 mL of yellow: blue = 10, yellow = 40, so the new ratio is 10 : 40 = \mathbf{1 : 4}.
  4. Parts = 3 + 1 = 4; each part = 8 \div 4 = 2 cups. Rice = 3 \times 2 = \mathbf{6} cups, urad dal = 1 \times 2 = \mathbf{2} cups.
  5. The new ratio cannot be given as a fixed number unless we know the tub size. The red amount is unchanged, but “another whole tub of yellow” depends on how much paint a tub holds. If each tub holds the same amount V, the original mix had \tfrac{2}{9}V red and \tfrac{7}{9}V yellow; adding V more yellow gives yellow = \tfrac{7}{9}V + V = \tfrac{16}{9}V, so red : yellow = \tfrac{2}{9}V : \tfrac{16}{9}V = \mathbf{2 : 16 = 1 : 8}.