Question

A man purchased sugar worth Rs. 400. He sold ${{\dfrac{3}{4}}^{th}}$ at a loss of 10% and the remaining at a gain of 10%. On the whole, he gets:
(a) a loss of 5%
(b) a gain of $5\dfrac{1}{2}\%$
(c) a loss of $5\dfrac{1}{19}\%$
(d) a gain of $5\dfrac{1}{19}\%$

Answer 1

Hint: We will first calculate how much is ${{\dfrac{3}{4}}^{th}}$ of Rs. 400. Then we will find the sold price (SP) at a loss of 10% and will be subtracting from the answer of ${{\dfrac{3}{4}}^{th}}$ . Then remaining i.e. $1-\dfrac{3}{4}=\dfrac{1}{4}$ we will be calculating this also. Similarly, we will calculate the sold price at a gain of 10% and will be adding it to $\dfrac{1}{4}$ amount. Then we will add both amounts of sold price to get the final SP. We will compare it with cost price (CP) i.e. Rs. 400 and check whether it is gain or loss. At last we will find loss or gain percentage using the formula $loss\%=\dfrac{loss}{CP}\times 100$ or $gain\%=\dfrac{gain}{CP}\times 100$ .

Complete step-by-step answer:
Here, we cost CP as Rs. 400. We will first calculate how much is ${{\dfrac{3}{4}}^{th}}$ of Rs. 400. Thus, on multiplying we get
$=400\times \dfrac{3}{4}$
On calculating, we get
$=Rs.300$ ………………….……….(1)
Now, let us say a whole is equal to 1. So, $\dfrac{3}{4}$ is already calculated. Remaining will be $1-\dfrac{3}{4}=\dfrac{1}{4}$ . Here, we will calculate how much is $\dfrac{1}{4}$ of Rs. 400. Thus, we will get
$=400\times \dfrac{1}{4}$
$=Rs.100$ ……………………….(2)
Now, it is given in question that ${{\dfrac{3}{4}}^{th}}$ is sold at a loss of 10%. So, we will find 10% of Rs. 300 which we already calculated in equation (1) and then will subtract it from Rs. 300. We will get
$=300\times 10\%$
On further simplification, we get
$=300\times \dfrac{10}{100}=Rs.30$
Now, subtracting Rs. 30 from Rs. 300 so, we will get SP as
$SP=Rs.300-Rs.30=Rs.270$ ……………………………….(3)
Also, it is told that remaining i.e. $\dfrac{1}{4}$ i.e. Rs.100 is sold at gain of 10%. So, calculating this, we will get
$=100\times 10\%$
On solving, we get
$=100\times \dfrac{10}{100}=Rs.10$
Now, as it the gain we will add this to Rs. 100 so we will get SP as
$SP=Rs.100+Rs.10=Rs.110$ ……………………………………..(4)
So, now on adding equation (3) and (4) we get total sold price SP as
$SP=Rs.270+Rs.110=Rs.380$ …………………………………….(5)
We know that is sold price is less than cost price i.e. $CP>SP$ then, it is a loss. So, Loss is calculated as CP minus SP.
 $loss=CP-SP=400-380=Rs.20$
Now, to find loss percentage we will use formula $loss\%=\dfrac{loss}{CP}\times 100$
On using this, we will get
$loss\%=\dfrac{20}{400}\times 100=5\%$
Thus, option (a) is correct.

Note: Be careful while calculating loss or gain because students forget to subtract or add in the main amount which results in wrong answers. Also, in finding the percentage of either loss of gain it should be on cost price CP and not on Sold price SP. If we find an SP, we will get an answer as $loss\%=\dfrac{20}{380}\times 100=5.263\%$ which is not in option and also the formula used is wrong. So, do not make this mistake.