Question

The ages of Han and Harry are in the ratio of 5:7. Four years from now, the ratio of their ages will be 3:4. Find their present ages.
(a) Present age of Han = 15 years, Present age of Harry = 21 years
(b) Present age of Han = 25 years, Present age of Harry = 35 years
(c) Present age of Han = 20 years, Present age of Harry = 28 years
(d) Present age of Han = 25 years, Present age of Harry = 30 years

Answer 1

Hint: To solve the question given above, we will assume that the present age of Han is x years and the present age of Harry is y years. Then we will take their ratio and we will equate it to \[\dfrac{5}{7}.\] Then we will find out their new ages i.e. ages after four years and we will equate it to \[\dfrac{3}{4}.\] Now, we will get a pair of the linear equations which we will solve by the method of substitution.

Complete step by step solution:
To start with, we will assume that the present age of Han is x years and the present age of Harry is y years. Now, it is given in the question that the ratio of their ages is 5:7. Thus, we will get the following equation.
\[\dfrac{x}{y}=\dfrac{5}{7}\]
On cross multiplication, we will get,
\[\Rightarrow 7x=5y\]
\[\Rightarrow 7x-5y=0.....\left( i \right)\]
Now, we will assume that their ages after four years will be x’ and y’. Thus, we can say that,
\[{{x}^{'}}=x+4.....\left( ii \right)\]
\[{{y}^{'}}=y+4.....\left( iii \right)\]
Now, we are given that the ratio of their new ages is 3:4. Thus, we will get the following equation,
\[\dfrac{{{x}^{'}}}{{{y}^{'}}}=\dfrac{3}{4}\]
Now, we will put the values of x’ and y’ from (ii) and (iii) to the above equation. Thus, we will get,
\[\dfrac{x+4}{y+4}=\dfrac{3}{4}\]
On cross multiplication, we will get,
\[\Rightarrow 4\left( x+4 \right)=3\left( y+4 \right)\]
\[\Rightarrow 4x+16=3y+12\]
\[\Rightarrow 4x-3y=12-16\]
\[\Rightarrow 3y-4x=4.....\left( iv \right)\]
Now, we have got a pair of linear equations in two variables. So, we will solve these equations by the method of substitution. In the substitution method, we write one variable in terms of other variables from one equation and we will put this value of the variable in the other equation. So, from equation (i), we have,
\[\Rightarrow 7x-5y=0\]
\[\Rightarrow 7x=5y\]
\[\Rightarrow y=\dfrac{7}{5}x.....\left( v \right)\]
Now, we will put the value of y from (v) to (iv). Thus, we will get,
\[\Rightarrow 3\left( \dfrac{7}{5}x \right)-4x=4\]
\[\Rightarrow \dfrac{21x}{5}-4x=4\]
\[\Rightarrow \dfrac{21x-20x}{5}=4\]
\[\Rightarrow \dfrac{x}{5}=4\]
\[\Rightarrow x=20\]
Now, we will put this value of x in the equation (v). Thus, we will get,
\[\Rightarrow y=\dfrac{7}{5}\left( 20 \right)\]
\[\Rightarrow y=7\times 4\]
\[\Rightarrow y=28\]
Therefore, the present age of Han is 20 years and the present age of Harry is 28 years.
Hence, the option (c) is the right answer.

Note: The linear equations in two variables which are formed during the solution can also be solved by graphical method. In this method, we will plot the graph of the two lines and then we will find the intersection point of these lines on the graph. This intersection point will be the values of x and y.