 QUESTION

# The ratio of the present ages of two brothers is $4:5$. After 10 years, the ratio of their ages would be $6:7$. Find the age of the elder brother after 15 years. (a) 35 years(b) 40 years (c) 50 years(d) 25 years

Hint: Assume that the age of one brother is ‘$x$’ and the age of the other brother is ‘$y$’. Form two linear equations in two variables according to the given conditions and solve for $x\text{ and }y$. Add 15 to the age of that brother whose age is found to be more, to get the answer.

Let us assume that the age of one of the brothers is ‘$x$’ years and that of the other is ‘$y$’ years. It is given that the ratio of their present ages is $4:5$. Therefore, mathematically,
$x:y=4:5$
Changing this into fraction, we get,
\begin{align} & \dfrac{x}{y}=\dfrac{4}{5} \\ & x=\dfrac{4y}{5}........................(i) \\ \end{align}
Now, after 10 years the age of the first brother will be $(x+10)$ years and that of the second brother will be $(y+10)$ years. it is given that after 10 years, the ratio of their ages will be $6:7$. Therefore, mathematically,
$(x+10):(y+10)=6:7$
Changing this into fraction, we get,
$\dfrac{x+10}{y+10}=\dfrac{6}{7}$
By cross-multiplication we have,
\begin{align} & 7x+70=6y+60 \\ & 6y-7x=10...................(ii) \\ \end{align}
Substituting the value of $x$ from equation (i) in equation (ii) we get,
\begin{align} & 6y-7\times \dfrac{4y}{5}=10 \\ & 6y-\dfrac{28y}{5}=10 \\ & \dfrac{30y-28y}{5}=10 \\ & \dfrac{2y}{5}=10 \\ & 2y=50 \\ & y=25 \\ \end{align}
Now, substituting the value of $y$ in equation (i), we get,
\begin{align} & x=\dfrac{4\times 25}{5} \\ & x=20 \\ \end{align}
Now, the age of the elder brother is 25 years. Therefore, his age after 15 years will be: $25+15=40$ years.
Hence, option (b) is the correct answer.

Note: One can also solve this question with the help of given options. Select any option and assume that it is correct. Now, satisfy all the conditions given in the question. If all the conditions are satisfied, then we can say that the selected option is correct. Otherwise if the conditions are not satisfied, then the selected option is wrong. But, this approach can only be executed when options are given.