Question

A dishonest dealer professes to sell his goods at cost price, but uses a false weight and thus gains 20%. For a kilogram he uses a weight of
(a)700g
(b)750g
(c)800g
(d)850g

Answer 1

Hint: Suppose the cost price of the goods as a variable by per kg of weight, and suppose the false weight as a variable as well. Now, calculate the exact price of the amount which he sold to get profit. Now, use the relation of percentage profit given as :-

Complete Step-by-Step solution:
% profit = Profit amount /original amount $\times $ 100

Let the dealer use x gm of weight instead 1 kg to gain 20% at his goods.
Let the original cost price of the goods is y Rs per 1 kg weight.
And the dealer professes to sell his goods at a price of y Rs per kg but he is using a false weight of x gm. It means the original price of the x gm goods is less than ‘y’.
So, we assumed that 1 kg has price of y Rs or we can convert 1 kg to 1000 gm and write that
Price of 1000 gm goods $=y$ Rs
So, price of 1 gm goods $=\dfrac{y}{1000}$ Rs
Hence, price of x gm goods $=\dfrac{y}{1000}\times x$ Rs
$=\dfrac{xy}{1000}$ Rs
It means that original price of x gm goods is $\dfrac{xy}{1000}$ Rs but dealer is selling it of y Rs and he is gaining 20% profit.
Now, we know the percentage profit of any item can be given by formula
% profit = Profit amount /original amount $\times $ 100 ……………………………..(i)
Now, we know that the profit amount is given by the difference of y Rs original price of 1 kg (dealer is selling x gm at y Rs) and $\dfrac{xy}{1000}$ Rs.
So, profit amount $=\left( y-\dfrac{xy}{1000} \right)$ Rs
Original amount $=y$ Rs.
Hence, we can calculate percentage profit as
Profit % $=\dfrac{y-\dfrac{xy}{1000}}{y}\times 100$
But, we have already given the percentage of 20%. So, we get
$20=\left( \dfrac{y-\dfrac{xy}{1000}}{y} \right)\times 100$
$20=\left( 1-\dfrac{x}{1000} \right)\times 100$
 $\dfrac{1}{5}=1-\dfrac{x}{1000}$
$\dfrac{1}{5}=\dfrac{1000-x}{1000}$
On cross multiplying the above equation we get
$1000=5000-5x$
$5x=4000$
$x=800Rs$
Hence, option (c) is correct.

Note: Direct formula for these kind of question can be given as
$\left( \dfrac{1-IW}{1} \right)\times 100=$ profit %
Where IW= Incorrect weight in kg.
This relation can be proved by the same approach done in the solution.
Cost price of the goods of 1 kg is the same amount at which the seller sold his/her goods. Profit has occurred because of the false weight. So, don’t confuse the cost price and the selling price of the goods.
Don’t put an amount of profit as ‘ $\dfrac{xy}{1000}-y$ ’ as $\dfrac{xy}{1000}$ is less than y. So, be careful while putting the value of profit amount in the percentage profit formula.