Question

A dealer buys dry fruits at the rate of Rs.100, Rs.80 and Rs.60 per kg. He bought them in the ratio 12:15:20 by weight. He in total gets 20% profit by selling the first two and at last he finds he has no gain no loss on selling the whole quantity he had. What is the percentage loss he suffered for the third quantity?
(a). 40%
(b). 20%
(c). 30%
(d). 50%

Answer 1

Hint: In this question, only the ratios of the weights in which the fruits are bought is given. Also, we have to find the percentage profit and thus the exact weights of the fruits is not necessary to solve this question. Instead we can assume the weight of bought fruits of the first type to be 12x kg so that the weights of the other fruits also become integers and then find the total cost price and selling price of the fruits. Using the information on the total profit, we can form an equation whose solution will give us the answer to the question.

Complete step-by-step solution -
Let the weight of the fruits bought of the first type be \[12x\] kg.
Then, as it is given that the ratio of the weights of the fruits is 12:15:20 by weight, the weight of the fruits bought of the second and third type should be \[15x\] kg and \[20x\] kg respectively.
Thus,
\[\begin{array}{*{35}{l}}
   Total\text{ }cost\text{ }price\text{ }of\text{ }the\text{ }fruits=\sum{\text{ }weight\text{ }of\text{ }the\text{ }fruits\text{ }per\text{ }kg\times \text{ }price\text{ }per\text{ }kg} \\
   ~12x\times 100+\text{ }15x\times 80+\text{ }20x\times 60=3600x....................(1.1) \\
\end{array}\]
where the $\sum{{}}$sign represents the sum over all the types of fruits.
The cost price of first two fruits would be
\[12x\times 100+15x\times 80=2400x..............(1.2)\]
We know the formula for percentage profit and percentage loss is
\[\text{Percentage profit}=\dfrac{\text{Selling Price}-\text{Cost Price}}{\text{Cost Price}}\times 100.............(1.3)\]
\[\text{Percentage loss}=\dfrac{\text{Cost Price}-\text{Selling Price}}{\text{Cost Price}}\times 100............(1.4)\]
It is given that the percentage profit of the first two fruits is 20%. Therefore, if the selling price of first two fruits is y, then using equations (1.2) and (1.3),
\[20=\dfrac{y-2400x}{2400x}\times 100\Rightarrow 20\times 24x+2400x=y\Rightarrow y=2880x..........(1.5)\]
Profit is equal to the selling price minus the cost price. As there is no profit in the total quantity, the total cost price should be equal to the total selling price. Thus,
\[\begin{array}{*{35}{l}}
   \text{Total selling price }=\text{ Total cost price }=\text{ }3600x \\
   \begin{align}
  & \Rightarrow y+\text{ selling price of third fruit}=3600x \\
 & \Rightarrow \text{selling price of third fruit}=3600x-2880x\text{ (from equation(1}\text{.5))} \\
 & \Rightarrow \text{selling price of third fruit}=720x.....................(1.6) \\
\end{align} \\
\end{array}\]
As 20x kg of third type of fruit was bought and its price was Rs.60 per kg,
$\text{Cost Price of third fruit}=60\times 20x=1200x..........(1.7)$
Now, using equation (1.4), we find that loss percentage of the third fruit is
$\text{Percentage Loss=}\dfrac{1200x-720x}{1200x}\times 100=0.4\times 100=40%$
Hence the answer is $40%$ which matches option (a). Thus the correct answer to this question is option(a).


Note: We should be careful to calculate the selling price of the third fruit and calculate its loss percentage and not simply state it as 20% as the profit percentage was 20% and net loss was zero. It is because the quantity of fruits bought and the cost price is different for each fruit and therefore the loss and profit should be calculated separately for each fruit.