Profit and Loss  Aptitude basics, practice questions, answers and explanations 
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Important formula and Equations

Gain= SP-CP
Loss= CP-SP
Gain Percentage= (Gain*100)/ CP
Loss Percentage= (Loss*100)/ CP
Selling Price=((100+Gain %)/100)*CP or ((100-Loss%)/100)*CP
Cost Price= (100*SP)/(100+Gain%) or (100*SP)/(100-Loss%)

When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by:

Loss %= (Common loss ans gain %)²/10 = (x/10)²

If a trader professes to sell his goods at cost price, but uses false weights, then

Gain%=((Error)/(True value)-(Error))*100%

Key Notes

When an article is sold at a profit of x%. If it would be sold for Rs.n less, there would be a loss of y%, then the cost price of the article CP=(n*100)/(x+y)

A man sells an article at a gain of x%. If it would have been sold for Rs.n more, there would have a profit of y%, then CP= (n*100)/(y-x)

A person brought two articles for Rs.n. On selling one article at x% profit and other at y% profit, he get the same selling price of each, then
CP of first article= Rs. (100+y)n/200+x+y

CP of second article= Rs. (100+x)n/ 200+x+y

When m articles are brought for Rs.n and n articles are sold for Rs.m and m>n, then profit%= ((m²-n²)*100)/n²
If A sells an article to B at a profit of r1 %, B sells it to C at a profit of r2 % and C sells it to D at a profit of r3 %, then, cost price of D= Cost Price of A (1+r1/100)(1+r2/100)(1+r3/100)

If A sells an article to B at a loss of r1 %, B sells it to C at a loss of r2 % and C sells it to D at a loss of r3 %, then, cost price of D= Cost Price of A (1-r1/100)(1-r2/100)(1-r3/100)

A dealer purchases a certain number of articles at x articles for a rupee and the same number at y articles for a rupee. He mixes them together and sells at z articles for a rupee.
Then his gain or loss %=([2xy- 1]/z(x+y))*100; according to positive or negative sign.

If P1 is rate gain w.r.t. selling price S1 and P2 is rate gain w.r.t. selling price S2
Then CP=(100/P1-P2)* difference between selling prices

If P1 is rate gain w.r.t. selling price S1 and P2 is rate loss w.r.t. selling price S2
Then CP= (100/ P1+P2) * difference between selling prices

When a man sells two things at the same price each and in this process his loss on first thing is x% and gain on second thing is x%, then in such a type question, there is always a loss and
Loss= 2*SP/((100/x)² -1)

When a man buys two things on equal price each and in those things one is sold on the profit of x% and another is sold on the loss of x%, then there is no loss or no gain percent.
A sells an article at a profit of r1 % to B and B again sells it to C at a profit of r2 %. If C pays Rs. P to B, then CP of the article for
A= Rs. 100*100*P/(100+r1)(100+r2)

When a shopkeeper on selling an article for Rs.n, gains as much percent as the cost price of it,then CP of the article 
                               

If there is loss in place of profit,
then CP of the article=  

If two articles are sold at the same price (i.e., the selling prices are equal) and the magnitude of percentage of profit x on one article is the same as the magnitude of percentage of loss x on the second article, then there is an overall loss and the percentage of loss is x²/100.

If a shopkeeper claims to sell the goods at cost price and gives x units less than the actual weight, then the profit percentage made by the shopkeeper is [x / actual weight – x] x 100.

In the above case, the error percentage = [x / actual weight] x 100

If two articles are bought for the same price (i.e., the cost prices are equal) and one is sold at a profit of p1% and the second is sold at a profit of p2%, then the overall percentage of profit is ((p1 + p2 )/2) x 100

If the selling price of m articles is equal to cost price of n articles, where m > n, then profit percentage is ((m – n )/m)x 100.
If m < n, then loss percentage = ((n – m)/m) x 100.

Discount

Discount% = Discount / Marked price * 100%

An article sold at selling price(SP1) at a loss of x% is to be sold at selling price(SP2) to gain y%, then SP2 = SP1(100 + y)/ (100-x)

If selling an object for Rs.x a person loses a certain sum and selling for Rs.y he gains the same amount, CP is given by CP = (x+y)/2.

When the price of an article is reduced by p% a man can buy x quantity of the article for Rs.y then
reduced price = 1/x ( y * p / 100) per unit.
original price = reduced price * 100 / (100 - p).

If the MP (marked price) of an article above CP is M% and after allowing a discount of d%, the gain is g%,
Then M% = d+g * 100% / 100 - d, and if there is a loss of l%, then M% = d-l * 100% / 100-d.

A person sells goods at a profit of x%. Had he sold it for Rs. X more, y% would have been gained. Then CP is given by Rs. X *100 / y-x.

A person sells goods at a loss of x%. Had he sold it for Rs. X more, he would have gained y% . Then CP is given by Rs. X * 100/ y+x.

When there are two successive profits of x% and y% the net gain% is given by: Net gain = [ x + y + { xy / 100 }]%.

When there are two successive losses of x% and y% the net loss% is given by: Net loss = [ - x - y + { xy / 100 }]%.10)

When there is a gain of x% and a loss of y% the net effect is given by: Net effect = [ x - y - { xy / 100}]%.

l. If d₁, d₂, d₃….. are percentages of successive discounts on a marked price MP, then the selling price SP = MP (1 – d₁/100) (1 – d₂/100) (1 – d₃/100)

2. If d₁, d₂, d₃…. are the percentages of successive discounts offered, then the effective discount is d% = 100[1- (1 – d₁/100) (1 – d₂/100) (1 – d₃/100)…]

3. If x and y are two successive discount percentages, then it is equivalent to a single discount percentage of x + y – xy/100.