# Concept 1

(This example illustrates the mistake if profit is calculated on SP instead of CP.)

**Illustration 1:**

A person claims to have earned a profit of 10%. But it was found that he had calculated this profit per cent on SP instead of CP. What is his actual profit?

Let SP = 100, Profit = 10, hence CP = SP - Profit = 90.

Actual profit = 10/90 = 11 1/9%.

Note that if the base is changed, we will get a different answer, hence it is important to always calculate profit on CP.

# Concept 2

To find SP when CP and Profit/Loss is given.

**Illustration 2:**

If I buy an article for Rs 300 what will its SP be if I make a (a) 10% profit, (b) 20% loss?

(a) CP = Rs 300; Profit = 10% of 300 = 30. Hence SP = 300 + 30 = 330

(b) CP = Rs 300; Loss = 20% of 300 = 60. Hence SP = 300 - 60 = 240.

Note that the same calculation can be done as follows: CP (1 + Profit %) = 300 (1 + 10%) = 300 (1.10) = 330.

In the second case, it becomes 300 (1 - 20%) = 300 (0.8) = 240.

This calculation eliminates one step so it is advisable to learn this.

# Concept 3

To find CP, when SP and Profit/Loss are given.

**Illustration 3:**

An article is sold for Rs 600. Find CP if it is sold for (a) 20% profit, and (b) 20% loss.

(a) SP = Rs 600;â€¨Let CP = 100. Then with 20% profit SP = 120.â€¨Using unitary method, we get: If SP is 120, CP = 100â€¨If SP is 600, CP = ?â€¨A quick calculation gives us that CP = 100/120 Ã— 600 = 500.

(b) Let CP = 100. Then with 20% loss, we get SP = 80.â€¨If SP is 80, CP = 100.â€¨If SP is 600, CP = ?â€¨Again by using unitary method, we get CP = 100/80 Ã— 600 = 750.

The above can also be done as follows: (a) 20% profit means CP + 20% of CP = 1.2 CP = 600, hence CP = 600/1.2 = 500.

In case of (b) we get CP - 20%(CP) = 0.8 CP = 600, hence CP = 600/0.8 = 750.

*The student should learn to do things in this way as it eliminates one step in the calculation.**â€¨*

**Illustration 4:**

The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is:

Let S.P. of x articles = Rs. 100= C.P. of 20 articles

C.P. of one article = Rs. 5â€¨

Profit = 25%â€¨

S.P. of one article = 6.25

S.P. of x articles = 6.25xâ€¨

6.25x = 100 => x = 100/6.25 = 16.

# Concept 4 - Expenses

So far we have considered only the SP and the CP. Sometimes additional information may be added by way of expenses. If we buy an article for Rs 1000 and we spend Rs 200 on transport, Rs 100 on octroi and Rs 100 as labour charges, then the expenses must be added to the CP to arrive at the actual CP. In this case, CP = Rs 1000+200+100+100 = Rs 1400.

**Illustration 5:**

John bought a car for a certain sum of money. He spent 10% of the cost of the car on repairs and sold the car for a profit of Rs 11,000. How much did he spend on repairs if he made a profit of 20%?

He makes 20% profit = 20% of CP.â€¨20% of CP = 11,000, hence CP = 55,000.â€¨

In this, there is 10% is for repairs.

Hence the CP is 55,000/1.1 = 50,000 and repairs are 10%, or Rs 5,000.

# Concept 5 - Marked Price

When we go to a shop, we look at the goods which are labelled with price tags. This is the Marked Price (MP) of the articles. However, this is not the Selling Price. The shopkeeper may give us a discount on the labelled price so the Selling Price is arrived at after deducting Discount on the Marked Price.

**Illustration 6:**

In one shop, an article is marked 75% above the cost price, but the purchaser is allowed a discount of 20% on the marked price. In another shop, a similar article is sold for Rs. 58 at a gain of 45%. What did the purchaser pay for this article in the first shop?

In the 1st shop:

Let C.P. = Rs. x

In the 2nd shop:â€¨S.P. = Rs. 58, Gain = 45%

âˆ´ C.P.= (58*100)/(100+45) = 40

âˆ´ S.P. in the 1st shop = 40 + 40% of 40 = Rs. 56.

# Concept 6 - Selling Two Things at Same SP and Same Gain/Loss%

If two items are sold each at Rs X, one at a gain of p% and the other at a loss of p%, then the two transactions have resulted in an overall loss of p2/100 %.

The absolute value of the loss = Rs (2 p2 X) / (1002 - p2)

**Illustration 7:**

A man sells two radios for Rs 924 each. On one he gains 12% and on the other he loses 12%. How much does he gain or lose on the whole?

Using the formula given above in the chapter, we can get the manâ€™s loss, which is (12/10)2 = 1.44%. If we do the sum by finding SP of the two articles, it becomes lengthy.

**Illustration 8:**

A man sold two houses for Rs 375890 each. On one he gains 15% and on the other he loses 15%. How much does he gain or lose in the whole transaction?

In such a type of question, there is *always a loss*. The SP is immaterial.â€¨We use the formula: Loss % = (Common loss and gain%/10)2. So loss % = (15/10)2 = 9/4 = 2.25%.

# Concept 7 - When Quantity is Reduced

**Illustration 9**

**:**

A grocer bought 10 Kg of apples for Rs. 81 out of which one Kg were found rotten. If he wishes to make a profit of 10%, then he should sell it at.....per Kg.

CP = Rs 81. He wants to make 10% profit. Sales price after profit of 10% = 81 + 10%(81) = 89.10. But 1 kg is rotten, so he has to sell 9 kg.â€¨Then sales price per kg = 89.10/9 = Rs 9.90 per kg.

# Concept 8 - Cash Discount

**Trade Discount:**

To attract customers it is a common practice to announce discount on the marked price of an article. The discount is always taken as a % of the marked price only unless otherwise specified.

**Illustration 10:**

Suppose the list price of an article be Rs 450. A discount of 5% on its list price is announced.

Hence the new selling price = (95*450)/100 = Rs 422.5

**Cash Discount:**

In addition to trade discount, the manufacturer may offer an additional discount called the Cash Discount if the buyer makes full payment within a certain specified time.

Cash Discount is usually offered on the net price (the price after subtracting discount from the marked price).

Therefore, Cash Price = Net Price - Cash Discountâ€¨

Cash discount is always calculated on net price, unless otherwise specified.

**Illustration 11**

**:**

A dealer offers a cash discount of 20% and still makes a profit of 20%, when he further allows 16 articles to a dozen to a particularly sticky bargainer. How much percent above the cost price were his wares listed?

Let the MP be 100, then SP for 16 articles is 80, after a 20% discount.

SP for 12 articles is 80 Ã— 12/16 = 60.â€¨This includes a profit of 20%, hence the CP must be 60/1.2 = 50. Marked price is 100, which is (100 â€“ 50)/50 Ã— 100 = 100% over the CP.

**Illustration 12**

**:**

If a dealer were to diminish the selling price of his wares by 12.5% he would double his sale making same profit as before. In what ratio would his profit diminish if he were to increase his selling price by 12.5% and thereby halve his sale?

Make a table as below:

As he makes the same profit as before, we get SP - CP = 2[(7/8)SP - CP]. Solving, we get CP = (3/4)SP.â€¨Now, we have new Selling Price = (9/8) SP and volume = 1/2.â€¨New profit = (1/2)[(9/8)SP - (3/4)SP] = (3/16)SP.

Old profit was (1/4)SP, so reduction in profit is 4:3, or (1).

# Concept 9 - Wrong Weight

**Using Wrong Weight**:*â€¨*

When a tradesman professes to sell at cost price, but uses a false weight, then the percentage profit earned = (100 Ã— error)/(true weight â€“ error).

**Illustration 13:**

A dishonest dealer professes to sell his goods at cost price, but uses a weight of 960 gm for a kg weight. Find his gain per cent.

To explain, let us say that he has 1000 gm to sell and the CP is Rs 1 per gm.â€¨When a customer asks for 1 kg, he charges CP, which is Rs 1000 but gives only 960 g. In this case, SP = 1000 and CP is only Rs 960.â€¨So he makes a profit of Rs 1000 â€“ 960 = Rs 40.â€¨Profit percentage = 40/960 Ã— 100 = 4 1/6%.â€¨We can also use the above formula to arrive at the same answer.

# Concept 10 - Successive Discounts

**Successive Discounts**:â€¨When a tradesman offers more than one discount to the customer, then start with 100 and keep applying the discounts to the decreased price.

**Illustration 14:**

A tradesman offers two successive discounts of 20% and 10%, which single discount is equal to these two successive discounts?

After the first discount of 20% the remaining price is 80 and after the second discount of 10%, the remaining part is 80 â€“ 8 = 72.

Hence single discount is 100 â€“ 72 = 28 i.e. 28%.