Question

Amrit , Ajay and Sudhanshu have some sweets with them. Amit gives half of his sweets to Ajay then Amrit and Ajay will have sweets in a 2:3 ratio. If Sudhanshu has one-fifth as many as Ajay
 (after Amrit gives his sweets to Ajay). Then what is the ratio of the number of sweets Amrit has to the number of sweets Sudhanshu has?

Answer 1

Hint: Here in the question we will proceed in two parts. Firstly using the ratio (2:3) between Amrit and Ajay, we will form a linear equation$\left( \dfrac{\dfrac{x}{2}}{y+\dfrac{x}{2}}=\dfrac{2}{3} \right)$ . Secondly using the relation between Ajay and Sudhanshu , we can easily find each share . Lastly getting each share, we can easily find the required ratio which is asked in the above question.

Complete step-by-step answer:
Let us consider that,
Amrit has ‘x’ sweets
Ajay has ‘y’ sweets
Sudhanshu has ‘z’ sweets
As it is given that Amrit gave half of his sweets to Ajay. So the remaining sweets he has now is $\dfrac{x}{2}$ .
So, the sweets that Ajay has now is $y+\dfrac{x}{2}$
Now, we will be using the ratio of sweets between Amrit and Ajay (after Amrit gives his sweets to Ajay) .
$\begin{align}
  & \Rightarrow \dfrac{\dfrac{x}{2}}{y+\dfrac{x}{2}}=\dfrac{2}{3} \\
 & \Rightarrow \dfrac{\dfrac{x}{2}}{\dfrac{2y+x}{2}}=\dfrac{2}{3} \\
 & \Rightarrow \dfrac{x}{2y+x}=\dfrac{2}{3} \\
 & \Rightarrow 3x=4y+2x \\
 & \Rightarrow y=\dfrac{x}{4}\cdots (i) \\
\end{align}$
Again, it is given that Sudhanshu has $\left( {{\dfrac{1}{5}}^{th}} \right)$ as many sweets as Ajay (after Amrit gives his sweets to Ajay). We have,
$\begin{align}
  & \Rightarrow z=\dfrac{1}{5}\left( y+\dfrac{x}{2} \right) \\
 & \Rightarrow z=\dfrac{1}{5}\left( \dfrac{x}{4}+\dfrac{x}{2} \right) \\
 & \\
\end{align}$
In the above we have used the equation (i)
$\begin{align}
  & \Rightarrow z=\dfrac{1}{5}\left( \dfrac{x+2x}{4} \right) \\
 & \Rightarrow z=\dfrac{3x}{20} \\
\end{align}$
Therefore the ratio of the number of sweets Amrit has (after Amrit gives his sweets to Ajay) to the number of sweets Sudhanshu has
$\begin{align}
  & =\dfrac{x}{2}:z \\
 & \\
\end{align}$
In the above we take $\dfrac{x}{2}$ for Amrit instead of ‘x’ because he has given half of his sweets to Ajay
$\begin{align}
  & =\dfrac{x}{2}:\dfrac{3x}{20} \\
 & =\dfrac{x}{2}\times \dfrac{20}{3x} \\
 & =\dfrac{10}{3} \\
 & =10:3 \\
\end{align}$
Hence is the ratio of the number of sweets Amrit has to the number of sweets Sudhanshu has i.e. x:z = 10:3 .

Note: In the question, it is clearly mentioned that Amrit has given half of his sweets to Ajay. So, when we find the ratio between them we must take $\dfrac{x}{2}$ as Amit’s share. Before solving any linear equation word problem, we must read the question carefully that what data is given and then form the linear equation as according to the given in question.