Dividing a quantity in a given ratio

Two friends put money together and buy some­thing. Later they share what they earn, but not equally. The one who put in more should get more. Shar­ing in a fixed ratio is the fair way to do it. In this les­son you will learn how to split any amount in a given ratio, into two parts or into three.

Shar­ing 30 in the ratio 2 : 3#

Sup­pose we must divide 30 in the ratio $2 : 3$.

The ratio $2 : 3$ means that for every 2 parts of the first share, the sec­ond share gets 3 parts. So there are $2 + 3 = 5$ equal parts in all.

Draw a line for 30 and cut it into 5 equal parts. Each part is $30 \div 5 = 6$.

The first share takes 2 parts: $2 \times 6 = 12$.

The sec­ond share takes 3 parts: $3 \times 6 = 18$.

Check: $12 + 18 = 30$, and $\displaystyle \frac{12}{18} = \frac{2}{3}$. Both are right.

The frac­tion way#

The first share is 2 parts out of 5, so it is $\displaystyle \frac{2}{5}$ of 30. The sec­ond share is $\displaystyle \frac{3}{5}$ of 30.

$$\displaystyle \frac{2}{2 + 3} \times 30 = \frac{2}{5} \times 30 = 12$$

$$\displaystyle \frac{3}{2 + 3} \times 30 = \frac{3}{5} \times 30 = 18$$

The k way#

Here is a sec­ond method. Call the two shares $2k$ and $3k$. Their ratio is $2k : 3k$, which is $2 : 3$, as it should be.

The shares add up to 30:

$$\begin{aligned} 2k + 3k &= 30 \\ 5k &= 30 \\ k &= 6 \end{aligned}$$

So the shares are $2k = 2 \times 6 = 12$ and $3k = 3 \times 6 = 18$.

Both meth­ods give the same answer. Use the one you like.

A rule for two parts#

If a num­ber $n$ is divided into two parts in the ratio $a : b$, then

$$\displaystyle \text{first part} = \frac{a}{a + b} \times n \qquad \text{second part} = \frac{b}{a + b} \times n$$

Why it works. Let the parts be $ak$ and $bk$. They add up to $n$:

$$\displaystyle \begin{aligned} ak + bk &= n \\ (a + b)k &= n \\ k &= \frac{n}{a + b} \end{aligned}$$

So the first part is $\displaystyle ak = \frac{an}{a + b}$ and the sec­ond part is $\displaystyle bk = \frac{bn}{a + b}$.

Exam­ple 1#

Divide ₹1500 between Sita and Ravi in the ratio $2 : 3$.

Total parts $= 2 + 3 = 5$.

$$\displaystyle \text{Sita's share} = \frac{2}{5} \times 1500 = 600$$

$$\displaystyle \text{Ravi's share} = \frac{3}{5} \times 1500 = 900$$

Sita gets ₹600 and Ravi gets ₹900. Check: $600 + 900 = 1500$.

A rule for three parts#

The same idea works for three parts. If a num­ber $n$ is divided into three parts in the ratio $a : b : c$, then

$$\displaystyle \text{first part} = \frac{a}{a + b + c} \times n$$

$$\displaystyle \text{second part} = \frac{b}{a + b + c} \times n$$

$$\displaystyle \text{third part} = \frac{c}{a + b + c} \times n$$

Why it works. Let the parts be $ak$, $bk$ and $ck$.

$$\displaystyle \begin{aligned} ak + bk + ck &= n \\ k(a + b + c) &= n \\ k &= \frac{n}{a + b + c} \end{aligned}$$

Mul­ti­ply $k$ by $a$, $b$ and $c$ to get the three parts.

Exam­ple 2#

Divide ₹2400 into three shares in the ratio $1 : 2 : 3$.

Total parts $= 1 + 2 + 3 = 6$.

$$\displaystyle \begin{aligned} \text{first share} &= \frac{1 \times 2400}{6} = 400 \\ \text{second share} &= \frac{2 \times 2400}{6} = 800 \\ \text{third share} &= \frac{3 \times 2400}{6} = 1200 \end{aligned}$$

The shares are ₹400, ₹800 and ₹1200. Check: $400 + 800 + 1200 = 2400$.

When we know some­thing else about the num­bers#

Some­times we are not told the total. We are told some other fact. The $k$ way still works.

Exam­ple 3#

The ratio of three num­bers is $3 : 4 : 5$ and the sum of their squares is 1250. Find the sum of the num­bers.

Let the num­bers be $3k$, $4k$ and $5k$.

The sum of their squares is 1250:

$$\begin{aligned} (3k)^2 + (4k)^2 + (5k)^2 &= 1250 \\ 9k^2 + 16k^2 + 25k^2 &= 1250 \\ 50k^2 &= 1250 \\ k^2 &= 25 \\ k &= 5 \end{aligned}$$

We take $k = 5$, because the num­bers here are pos­i­tive.

The num­bers are 15, 20 and 25. Their sum is

$$3k + 4k + 5k = 12k = 12 \times 5 = 60$$

Check: $15^2 + 20^2 + 25^2 = 225 + 400 + 625 = 1250$. Cor­rect.

Exam­ple 4#

Two num­bers are in the ratio $3 : 7$ and their dif­fer­ence is 36. Find the num­bers.

Let the num­bers be $3k$ and $7k$. The big­ger one minus the smaller one is 36.

$$\begin{aligned} 7k - 3k &= 36 \\ 4k &= 36 \\ k &= 9 \end{aligned}$$

The num­bers are $3 \times 9 = 27$ and $7 \times 9 = 63$. Check: $63 - 27 = 36$.

Remem­ber#

• Add the terms of the ratio to find the total num­ber of parts.
• Each share is (its own term) over (total parts), times the amount.
• Always check that the shares add up to the amount.

Prac­tice#

1. Divide 40 in the ratio $3 : 5$.
2. Divide ₹720 between Anil and Sunil in the ratio $5 : 7$.
3. Divide 84 in the ratio $1 : 2 : 4$.
4. Divide ₹3600 among three chil­dren in the ratio $2 : 3 : 4$.
5. A rope 45 m long is cut into two pieces in the ratio $4 : 5$. Find the length of each piece.
6. Two num­bers are in the ratio $5 : 8$ and their sum is 91. Find the num­bers.
7. The angles of a tri­an­gle are in the ratio $2 : 3 : 4$. Their sum is 180 degrees. Find each angle.
8. Two num­bers are in the ratio $4 : 9$ and their dif­fer­ence is 30. Find the num­bers.
9. Three num­bers are in the ratio $2 : 3 : 4$ and the sum of their squares is 1856. Find the num­bers.

Answers#