Question

Toshiba bought 100 hens for Rs.8000 and sold 20 of these at a gain of 5%. At what gain percent she must sell the remaining hens so as to gain 20% on the whole?

Answer 1

Hint: In this question, we are given a cost price of 100 hens and gain percentage on selling price of 20 hens. Using this, we have to tell gain % on 80 hens such that the gain percentage on selling price of all hens (100 hens) is 20%. For this, we will first calculate the cost price of one hen and then find the cost price of 20 hens. Using gain percentage and cost price of 20 hens in the formula $SP=CP\dfrac{\left( 100+\text{Gain}\% \right)}{100}$, we will find a selling price of 20 hens. Then we will find the cost price of 80 hens and gain on 100 hens in terms of SP of 80 hens using formula: Gain = SP - CP. At last, we will apply the formula $\text{Gain}\%=\dfrac{\text{Gain}}{CP}\times 100$ and calculate SP of 80 hens. Using that, we will find gain and then gain % of 80 hens.

Complete step by step answer:
Here, we are given a cost price of 100 hens as Rs.8000. Therefore, the cost price of one hen will be $Rs.\dfrac{8000}{100}=Rs.80$. Now, let us find the cost price of 20 hens.
Cost price of 20 hens will be $Rs.\left( 20\times 80 \right)=Rs.1600$.
We are given gain % on 20 hens as 5%. Using cost price and gain % let us find a selling price of 20 hens.
We know, $SP=CP\dfrac{\left( 100+\text{Gain}\% \right)}{100}$.
Putting values of CP and gain % we get:
\[\begin{align}
  & SP=Rs.1600\dfrac{\left( 100+5 \right)}{100} \\
 & \Rightarrow Rs.1600\left( \dfrac{105}{100} \right) \\
 & \Rightarrow Rs.\left( 16\times 105 \right) \\
 & \Rightarrow Rs.1680 \\
\end{align}\]
Hence, the selling price of 20 hens is Rs.1680.
Now, let us find the cost price of 80 hens.
Cost price of 80 hens will be $Rs.\left( 80\times 80 \right)=Rs.6400$.
We know that, Gain = SP - CP.
So, Gain on 80 hens = SP of 80 hens - CP of 80 hens.
Gain on 80 hens = SP of 80 hens - Rs.6400.
Gain on 80 hens = Rs.(SP of 80 hens - 6400).
Also, gain on 100 hens will be equal to sum of gain on 80 hens and gain on 20 hens
So, Gain of 100 hens = Gain of 80 hens + Gain of 20 hens.
Gain of 100 hens = Rs.(SP of 80 hens - 6400) + Gain of 20 hens.
Let us calculate gain on 20 hens.
Now, Gain = SP - CP. Therefore,
Gain of 20 hens = SP of 20 hens - CP of 20 hens.
$\Rightarrow 1680-1600=Rs.80$.
Therefore, gain on 100 hens becomes equal to
\[\begin{align}
  & \Rightarrow \text{Rs}.\left( \text{SP of 8}0\text{ hens}-\text{64}00 \right)+Rs.80 \\
 & \Rightarrow \text{Rs}.\left( \text{SP of 80 hens}-\text{64}00+80 \right) \\
 & \Rightarrow \text{Rs}.\left( \text{SP of 8}0\text{ hens} \right)-Rs.6320 \\
\end{align}\]
Hence, gain on 100 hens is (SP of 80 hens) - Rs.6320.
We know, $\text{Gain}\%=\dfrac{\text{Gain}}{CP}\times 100$.
So, $\text{Gain}\%\text{ of 100 hens}=\dfrac{\text{Gain on 100 hens}}{\text{CP of 100 hens}}\times 100$.
We require gain % on 100 hens as Rs.20. So, equation becomes
\[\begin{align}
  & \Rightarrow 20=\dfrac{\text{SP of 80 hens}-6320}{8000}\times 100 \\
 & \Rightarrow 20=\dfrac{\text{SP of 80 hens}-6320}{80} \\
 & \Rightarrow 1600=\text{SP of 80 hens}-6320 \\
 & \Rightarrow \text{SP of 80 hens=1600+}6320 \\
 & \Rightarrow \text{SP of 80 hens}=Rs.7920 \\
\end{align}\]
Hence, selling price of 80 hens is Rs.7920.
As we know, Gain = SP - CP.
So, Gain on 80 hens = SP of 80 hens - CP of 80 hens
$\Rightarrow Rs.7920-6400=Rs.1520$.
Now, we know $\text{Gain}\%=\dfrac{\text{Gain}}{CP}\times 100$. So,
\[\begin{align}
  & \text{Gain}\%\text{ of 80 hens}=\dfrac{\text{Gain on 80 hens}}{\text{CP of 80 hens}}\times 100 \\
 & \Rightarrow \dfrac{1520}{6400}\times 100 \\
 & \Rightarrow 23.75\% \\
\end{align}\].

Hence, gain % of 80 hens should be 23.75% so that gain % of 100 hens become 20%.

Note: Students can get confused between SP, CP, Gain, Gain % of different number of hens. So they should take care while using formula. Students should know that, gain and gain % are different. Gain is calculated by SP - CP and gain % is calculated by $\dfrac{\text{Gain}}{CP}\times 100$.