# Discount

## Terms

### Marked price (MP)

Marked price or mark up price (MP) is the price which the shopkeeper fixes in anticipation of some discount being asked by customer.

Marked price (M.P.) = C.P. + markup

### Discount (D)

Discount means reduction of marked price to sell at a lower rate. It is calculated on the basis of marked price (M.P.)

SP = MP – Discount = MP (1 – Discount%/100)

[Note that (1 – Discount%/100) is the multiplying factor.]

If no discount is given, then
So, SP = MP

### Discount percentage

Profit or Loss percentage is always calculated on the cost price. Whereas, Discount percentage is calculated on marked price (or list price).

Discount percent = (Discount / MP) x 100

## Finding net profit/loss percentage

The shopkeeper buys at CP and then marks the item up by m% (of CP) to get MP. After that he gives a discount of d% (of MP) to get SP.

So, if a shopkeeper marks the products at m% above the cost price and gives the customer a discount of d%, then

final profit or loss % = m − 𝑑 − $\frac{(m × 𝑑)}{100}$% (successive percentage changes formula)

In terms of multiplying factors, $MF_{loss/profit}$ = $MF_{mark-up}$ × $MF_{discount}$ = (MP/CP) × (SP/MP) = SP/CP
##### Q. If a businessman marks up his product by 15% over the cost price and thereafter offers a discount of 20%, then find out his profit/loss percentage.

Explanations :

Explanation 1: Percentage Method

Let initial C.P. be Rs 100. Then, M.P. = 100 + 15% of 100 = Rs. 115

Now, S.P. = 115 – 20% of 115 = 115 - 23 = Rs. 92

Note: We can also calculate S.P. as follows: S.P. = (80/100) × 115 = Rs. 92

So, loss percentage = (loss/C.P.) × 100 = (8/100) × 100 = 8%

Explanation 2: Formula Method (using successive percentage changes formula)

So, if a shopkeeper marks the products at m% above the cost price and gives the customer a discount of d%, then

final profit or loss % = m − 𝑑 − $\frac{(m × 𝑑)}{100}$% = 15 − 20 − $\frac{(15 × 20)}{100}$% = -5 – 3 = -8%
(minus sign denotes that there has been a loss)

Explanation 3: Multiplying Factor Method

In terms of multiplying factors, $MF_{loss/profit}$ = $MF_{mark-up}$ × $MF_{discount}$ = 1.15 × 0.80 = 0.92

So, it means that SP/CP = 0.92

Hence, loss percentage = (1 – 0.92) × 100 = 8%

Explanation 4: Fraction Method

15% = 3/20. So, if CP = 20, then MP = 20 + 3 = 23

20% = 1/5. So, if MP = 5, then SP = 5 – 1 = 4

But we know MP is 23, so SP = (4/5) × 23

So, if CP = 20, then SP = (4/5) × 23

Therefore, if CP = 100, then SP = (4/5) × 23 × (100/20) = 92

So, loss percentage = 8%

Instead of doing all this, we can shorten our calculation by doing this:

15% = 3/20, so MP/CP = 23/20
20% = 1/5, so SP/MP = 4/5
So, SP/CP = (23/20) × (4/5) = 92/100

Hence, loss percentage = 8%

##### Q. A shopkeeper gives 30% discount on the clothes he sell, but still manages to make a profit of 5%. What must be the mark up percentage?

Explanations :

Explanation 1: Percentage Method

Let CP = Rs. 100

As he makes a profit of 5%, so SP = 100 + 5% of 100 = Rs. 105

As 30% discount was given, we can say that 70% is equivalent to Rs. 105
Hence, 100% will be equivalent to 105 × (100/70) = Rs. 150

So, mark-up percentage = (50/100) × 100 = 50%

Explanation 2: Fraction Method

30% discount = 3/10, so SP/MP = 7/10
5% profit = 1/20, so SP/CP = 21/20
So, MP/CP = (10/7) × (21/20) = 3/2 = 1.5

Hence, mark-up percentage = 50%

Let C.P. = Rs. 100 and the mark up percentage be x%.
Hence, M.P. = Rs. 100 + x
And S.P. = 70% of (100 + x) = 0.70 (100 + x)

As he makes a profit of 5%, his S.P. must be Rs. 105.

So, 0.70 (100 + x) = 105
or 70 + 0.7x = 105
or 0.7x = 35
or x = 50%

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