Question

The ages of Sona and Sonali are in the ratio of 5:3. Five years later, the ratio of their ages will be 10:7. Find their present ages.

Answer 1

Hint: To solve this question, we will assume that the present age of Sona is x years and the present age of Sonali is y years. Then we will take their ratio and equate it to 5:3. Then we will find out their new ages by adding 5 to their present ages. And we will equate it to 10:7. Now, we will get a pair of linear equations in two variables, which we will solve with the method of elimination.

Complete step by step solution:
To start with, we will assume that the present age of Sona is x years and the present age of Sonali is y years. Now, it is given in the question that the ratio of their ages is 5:3. Thus, we will get the following equation,
\[\dfrac{x}{y}=\dfrac{5}{3}\]
On cross multiplication, we will get,
\[\Rightarrow 3x=5y\]
\[\Rightarrow 3x-5y=0....\left( i \right)\]
Now, let us assume that their ages after five years are x’ and y’ respectively. Thus, we can say that,
\[{{x}^{'}}=x+5....\left( ii \right)\]
\[{{y}^{'}}=y+5....\left( iii \right)\]
Now, we are given that the ratio of their new ages is 10:7. Thus, we will get the following equation,
\[\dfrac{{{x}^{'}}}{{{y}^{'}}}=\dfrac{10}{7}\]
On cross multiplication, we will get,
\[\Rightarrow 7{{x}^{'}}=10{{y}^{'}}\]
\[\Rightarrow 7{{x}^{'}}-10{{y}^{'}}=0....\left( iv \right)\]
Now, we will put the values of x’ and y’ from (ii) and (iii) into (iv). Thus, we will get,
\[\Rightarrow 7\left( x+5 \right)-10\left( y+5 \right)=0\]
\[\Rightarrow 7x+35-10y-50=0\]
\[\Rightarrow 7x-10y-15=0\]
\[\Rightarrow 7x-10y=15....\left( v \right)\]
Now, we have got a pair of linear equations in two variables. So, we will solve these equations by the method of elimination. In the elimination method, we multiply one equation with a non – zero constant such when it is added or subtracted from another equation, one variable is eliminated. So, we will apply this method now.
For this, we will multiply equation (i) with – 2. Thus, we will get,
\[\Rightarrow \left( -2 \right)3x-\left( -2 \right)5y=0\left( -2 \right)\]
\[\Rightarrow 10y-6x=0.....\left( vi \right)\]
Now, we will add the equations (v) and (vi). Thus, we will get,
\[\Rightarrow \left( 7x-10y \right)+\left( 10y-6x \right)=15+0\]
\[\Rightarrow 7x-10y+10y-6x=15\]
\[\Rightarrow x=15....\left( vii \right)\]
Now, we will put this value of x from (vii) to (i). After doing this, we will get,
\[\Rightarrow 3\left( 15 \right)-5\left( y \right)=0\]
\[\Rightarrow 5y=45\]
\[\Rightarrow y=9\]
Therefore, the present age of Sona is 15 years and the present age of Sonali is 9 years.

Note: The linear equations in two variables which are formed during the solution can also be solved with the help of the substitution method. In this method, we will put the values of one variable in terms of other variables from one equation to other equations. Thus, we have,
\[3x-5y=0\]
\[\Rightarrow \dfrac{3}{5}x-y=0\]
\[\Rightarrow y=\dfrac{3}{5}x\]
Now, we will put this value of y in equation (v).
\[7x-10\left( \dfrac{3}{5}x \right)=15\]
\[\Rightarrow 7x-6x=15\]
\[\Rightarrow x=15\]
Now, the value of y is,
\[y=\dfrac{3}{5}x\]
\[\Rightarrow y=\dfrac{3}{5}\times 15\]
\[\Rightarrow y=9\]