To understand the concept of profit and loss, we first have to understand the concept of ‘Percentage’. Percentage, which is clear by name, implies “for every hundred”. This concept is actually developed to make the comparison of fractions easier by equalizing the denominators of all fractions to hundred.

e.g. 1.  4/9 can be represented as,

⇒ 4/9 × 100 = 44.44%

It means, 4/9 is equal to 44.44/100.

Now, any Percentage can be expressed as a decimal fraction by dividing the percentage figure by 100 and any decimal fraction can be converted to percentage by multiplying it by 100.

e.g. 2.  40% is same as 40/100 = 0.4

 PERCENTAGE INCREASE or DECREASE:

Now, there is a most important concept, to understand profit and loss, called PERCENTAGE INCREASE or DECREASE.

PERCENTAGE INCREASE or DECREASE in a quantity is the ratio, expressed in percentage, of the actual INCREASE or DECREASE in the quantity to the original amount of the quantity.

i.e.  PERCENTAGE INCREASE = (Actual increase/Original quantity) × 100

PERCENTAGE DECREASE = (Actual decrease/Original quantity) × 100

e.g. 3.  If the consumption of rice by a family is increased from 60 kg/month to 75 kg/month, then the percentage increase in rice consumption is calculated as follows:

⇒ Actual increase = 75 – 60 = 15 kg/month

⇒ Percentage increase = (actual increase/quantity at the beginning) × 100

⇒ 15/60 × 100 = 25%

 PROFIT/LOSS:

The main parts of this topic are,

SELLING PRICE (S.P.) = the price at which a product was sold.

COST PRICE (C.P.) = the price at which a product was purchased.

MARKED PRICE (M.P.) = the price which is written on the product (we’ll discuss it later)

Now, the computation is done as follows:

Profit = Sale Price – Cost Price = S.P. – C.P.

Percentage Profit = (S.P. – C.P.)/C.P.  × 100 = (Profit/C.P.) × 100

Loss = Cost Price – Sale Price = C.P. – S.P.

Percentage Loss = (Loss/C.P.) × 100

e.g. 4.  Amar brought a pair of shoes for Rs. 250 and sold it for Rs. 300. Find the profit percentage.

Ans:-  Here,

           S.P. = Rs. 300, C.P. = Rs. 250

           ⇒ Profit = S.P. – C.P. = 300 – 250 = Rs. 50

           ⇒ Profit% = (50/250) × 100 = 20%

We can take a more example to understand it clearly,

e.g. 5.  Arun calculated her profit/loss percentage on selling prices. Find her actual profit/loss percentage if she calculated

(i) her profit percentage to be 25%

(ii) her loss percentage to be 25%

Ans:- We know that profit percentage or loss percentage is calculated on cost price. So first we have to find the cost price and then actual profit/loss percentage.

Let her selling price be Rs. 100 (to ease our calculation, otherwise we could take x also)

(i) Profit = Rs. 25 = S.P. – C.P.

⇒ C.P. = S.P. – Profit = 100 – 25 = Rs. 75

Now, actual profit percentage

⇒ (25/75) × 100 = 33.33%

(ii) Loss = Rs. 25 = C.P. – S.P.

⇒ C.P. = Loss + S.P. = 25 + 100 = Rs. 125

Now, actual loss percentage

⇒ (25/125) × 100 = 20%

Now, let’s get familiar with MARKED PRICE with an example,

e.g. 6.  Ajay marked his goods 40% above his cost price. He then gave a discount of 20%. Find his profit percentage.

Ans:- Let the cost price be Rs. 100

⇒ Marked Price = Actual Price + increased percentage = 100 + [(40/100) × 100] = 140

⇒ Selling Price = Marked Price × (100 – Discount %) /100 = 140 × 80/100 = 112

⇒ Profit % = [(112 – 100)/100] × 100 = 12%

Let’s try something productive to ease our calculation.

MULTIPLYING FACTOR:

It’s way too lengthy to take a quantity and then convert that into 100 and then check the percentage change and then convert back into original quantity. So, to ease our calculation, let’s understand this thing.

If the increase on a value of 350 is 15%, the new quantity is 1.15 × 350 = 402.5

(Here 1.15 = 1 + 0.15 = 100% + 15%)

It means we can directly calculate the new value of quantity instead of calculating the actual increase/decrease and then add to/subtracting from original quantity.

So, this 1.5, in case of pervious example, is called Multiplying Factor.

In general,

M.F. (in case of % increase) = (p/100) + 1

M.F. (in case of % decrease) = 1 – (p/100)

(Where p is percentage change given)

So,

⇒ New quantity = M.F. × Original quantity

In another words,

⇒ Selling Price = M.F. × Cost price

e.g. 7.  If the decrease on the value of 350 is 20% the new quantity is 0.8 × 350 = 280

[Here 0.8 = 1 – 0.2 = 1 – (20/100)]

Now, let’s try to solve example 6 again.

Ans:- Let the cost price is Rs. 100

⇒ Marked Price = 1.4 × 100 = Rs. 140

⇒ Selling Price = 0.8 × 140 = Rs. 112

⇒ Profit% = (112 – 100)/100 × 100 = 12%

SUCCESSIVE PERCENTAGE CHANGE:

If the percentage change is done in the quantity by two or more times, there is a bit too hard calculation to find the value. Let us take an example

e.g. 8.  A vegetable seller increases the price of onions by 30%. But after seeing that the demand of onions is high, he further increases the price of onions by 20%. If the initial selling price of onions is Rs. 40/kg, find the new selling price of onions.

Ans:- Initial price of onions = Rs. 40

⇒ after first increment, price of onions = 1.3 × 40 = Rs. 52

⇒ after second increment, price of onions = 1.2 × 52 = Rs. 62.4

So, the new selling price of onions is Rs. 62.4/kg

It is looking an easy question but it can be tough when the percentage change will be given in some complicated numbers like 47% and 23%

So, to escape this two times calculation, we can use this formula,

If a and b are two successive percentage changes, then,

Final change = a + b + (ab/100)

Let’s take the previous example,

Here a = 30% and b = 20%

⇒ Final change = 30 + 20 + (30 × 20)/100 = 56%

⇒ Final value of onions = 1.56 × 40 = Rs. 62.4/kg

Let’s take another example,

e.g. 9.  Chetan sold his goods after announcing two successive discounts of 30% each. Find his effective discount percentage.

Ans:- Here a = -30% (it is discount, not increase) and also b = -30%

⇒ Final change = – 30 – 30 + (-30 × -30)/100 = -51%

⇒ Effective discount percentage is 51%

 Here are some other formulas, which you can apply and reach directly to the answer and/or save your time without going through actual lengthy and time taking process.

• If the value of an item up/down by x%, the percentage reduction/increment to be now made to bring it back to original level is [100x/(100 ± x)]%.

Same formula we can use to find if A is x% more/less than B, then B is how much less/more than A.

e.g. 10.  If A is 20% more than B, then B is [(100 × 20)/(100 – 20)]% = 25% less than A.

• Given the cost price (CP) and the profit/loss percentage p%, the selling price will be given by

SP = CP × (100 ± p)/100

• Given the selling price (SP) and the profit/loss percentage p%, the cost price will be given by

CP = SP × 100/(100 ± p)

• When two articles are SOLD at the same price (same SP) such that there is a PROFIT of p% on one article and a LOSS of p% on other, then, irrespective of what the SP actually is, the net result of transaction is LOSS. This percentage is given by,

Loss percentage = (common profit or loss)2/100 = p2/100

e.g. 11.  Dheeraj sold two harmoniums, one at 10% profit and the other at 10% loss. Find his overall profit/loss percentage if he sold both the sets at the same price.

Ans:- (Without formula)

Let the SP is Rs. 100

Also, Multiplying factor × CP = SP

⇒ 1.1 × CP of first harmonium = 100

⇒ CP of first harmonium = 100/1.1 = Rs. 90.909

⇒ 0.9 × CP of second harmonium = 100

⇒ CP of second harmonium = 100/0.9 = Rs. 111.111

⇒ Total Cost Price = 90.909 + 111.111 = Rs. 202.02

⇒ Total Selling Price = 100 + 100 = Rs. 200

Clearly, here is a loss.

⇒ Net loss percentage = (202.02 – 200)/200 × 100 = 1.01%

(With formula)

We know that irrespective of what SP actually is, net result is loss.

⇒ Loss percentage = (10 × 10)/100 = 1%

• In the case of successive discounts, if the successive discounts are p%, q% and r%, on a product whose selling price is S.P., then the effective price after all the discounts is given by

Discounted price = S.P. × [(100 – p)(100 – q)(100 – r)/(100 × 100 × 100)]