Question
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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.3:5
b.2:5
c.2:3
d.3:4

Answer Verified Verified
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

Complete step by step answer:

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 $3:5$
The answer is option a.

Note: When the ratio of two quantities is given like the ages of two persons is 2:5. That does not mean that the ages are 2 and 5. Most of the students make a mistake here.
After finding the ages we just can’t stop by leaving the ratio at 15:25. We need to reduce it to its simpler form in order to get full marks.