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Question: A person having goods for Rs. 400 sells half of it at a gain of 5%. At what gain% must he sell the r...

A person having goods for Rs. 400 sells half of it at a gain of 5%. At what gain% must he sell the remainder so, as to gain 20% on the whole?
(a) 35%
(b) 40%
(c) 30%
(d) 25%

Explanation

Solution

First, we will find the whole gain amount that is (SP) 20% on cost price (CP) Rs. 400 using the formula SP=CP+gain%⋅CPSP=CP+gain\%\cdot CP . The, we will find SP of 5% on half the value of CP i.e. Rs. 200. After getting this we will find the difference between both the SP we got. And then we will find gain% on the difference price we got over the remaining cost price i.e. Rs. 200 using the formula gain%=gain priceCP×100gain\%=\dfrac{gain\text{ }price}{CP}\times 100 . Thus, we will get the answer.

Complete step-by-step answer:
Here, it is said that a person gets 20% gain in whole on the price of goods i.e. cost price (CP) Rs. 400. So, we will find that gain amount which is calculated by the formula given as sale price (SP) SP=CP+gain%⋅CPSP=CP+gain\%\cdot CP .
So, we will get as
SP=CP+gain%⋅CPSP=CP+gain\%\cdot CP
On substituting the values, we get
SP=400+20%⋅400SP=400+20\%\cdot 400
SP=400+20100⋅400SP=400+\dfrac{20}{100}\cdot 400
On further solving, we will get
SP=400+20⋅4=400+80=Rs.480SP=400+20\cdot 4=400+80=Rs.480 ……………………………….(1)
Now, it is said that a person gets 5% gain if he sells at half price of CP. So, we can calculate SP using the same formula SP=CP+gain%⋅CPSP=CP+gain\%\cdot CP where CP will be half of Rs. 400 i.e. Rs. 200.
So, we will get
SP=200+5%⋅200SP=200+5\%\cdot 200
On further solving, we get
SP=200+5100⋅200SP=200+\dfrac{5}{100}\cdot 200
SP=200+5⋅2=200+10=Rs.210SP=200+5\cdot 2=200+10=Rs.210 …………………………………..(2)
Now, the difference between the selling price SP is =480−210=Rs.270=480-210=Rs.270 .
Now, in the remaining half i.e. Rs. 200 a person must gain Rs. 70 more to get a total amount of Rs. 270. So, we get gain% as gain%=gain priceCP×100gain\%=\dfrac{gain\text{ }price}{CP}\times 100 .
=70200×100=35%=\dfrac{70}{200}\times 100=35\%
Hence, option (a) is the correct answer.

Note: Do not make mistakes in finding the gain% over the remaining cost price. Students generally do not understand what to consider in cost price to find gain%. Instead of getting cost price CP as Rs. 200, students take CP as Rs. 270 and using the formula gain%=gain priceCP×100gain\%=\dfrac{gain\text{ }price}{CP}\times 100 we will get answer as
=70270×100=25.9%=\dfrac{70}{270}\times 100=25.9\% .
So, do not assume to be 25% and select option (d). This is the wrong answer. Be careful while taking CP and to avoid mistakes.