Question

# The ratio of the present ages of two brothers is 1:2 and 5 years back the ratio was 1:3. What will be the ratio of their ages after 5 years?a.3:5b.2:5c.2:3d.3:4

Hint: We need to assume the present ages to be $x,2x$, and from this, we can say that the ages of the brothers before 5 years are $x - 5$ and $2x - 5$. Find the ratio between these two and equate with a given ratio and we will get a linear equation in x. solving which we can get the present age and adding 5 to the present age gives the age after five years

Step 1:It is given that the ratio of the present ages of two brothers is $1:2$
From this, let us assume the ages to be $x,2x$
Step 2:Five years before the ages of the brothers will be $x - 5$ and $2x - 5$ respectively.
Let the ratio of these ages be $\dfrac{{x - 5}}{{2x - 5}}$. …………..(1)
Step 3: It is also given that, five years back the ratio was $1:3$.
This can be written as $\dfrac{1}{3}$………(2)
Step 4:Now by equating (1) and(2), we get
$\Rightarrow \dfrac{{x - 5}}{{2x - 5}} = \dfrac{1}{3}$
By cross multiplying we get,
$\begin{gathered} \Rightarrow 3(x - 5) = 1(2x - 5) \\ \Rightarrow 3x - 15 = 2x - 5 \\ \end{gathered}$
Lets bring the terms to one side and the constants to the other side
$\begin{gathered} \Rightarrow 3x - 2x = 15 - 5 \\ \Rightarrow x = 10 \\ \end{gathered}$
Step 5:Now, we have that $x = 10$, which is the present age of one of the brothers.
From this,lets calculate the age of the other brother (i.e.) $2x = 2(10) = 20$
Step 6: We are asked to find the ratio of the ages after five years.
The ages of the brothers after five years are $x + 5$ and $2x + 5$
$\begin{gathered} \Rightarrow x + 5 = 10 + 5 = 15 \\ \Rightarrow 2x + 5 = 2(10) + 5 = 20 + 5 = 25 \\ \end{gathered}$
Step 7: Now lets find the ratio of the ages
$\Rightarrow \dfrac{{15}}{{25}}$
And now we need to simply the above fraction to the simplest form
$\Rightarrow \dfrac{{5*3}}{{5*5}} = \dfrac{3}{5}$
Therefore the ratio of ages after five years is $3:5$