Consider a situation where you are offered the same item by two shopkeepers at the same price. But first offering a discount of 45% and the other at successive discounts of 20%, 20%, and 10%.

Which one will you choose? Which of the above will be a better deal for you?

To understand this, let’s first understand the meaning of successive discounts and how it is calculated using the successive discount formula.

## Terms Related to Successive Discount

Before moving further, let’s understand the following related terms.

**Marked Price / List Price:** The price on the label of an article/product is called the marked price or list price. This is the price at which the product is intended to be sold.

*However, there can be some discount given on a list price and then the actual selling price of the product will be less than the marked price.*

**Discount:** A discount is a deduction from the usual price of something. To discount means to deduct an amount from the price.

(Sale Price) = (Marked Price) – Discount

For example, if the marked price of an item is ₹150 and the discount is ₹20, then the sale price will be 150 – 20 = ₹130.

Often discounts are represented in terms of percentage. For example 20%, 40%, etc. In such cases, the discount amount is computed on the marked price or list price of an item.

For example, if the marked price of an item is ₹150 and the discount is 20%, then first of all discount amount is calculated and then the sale price is calculated.

Discount = (Marked Price) $\times$ (Discount Percent) = $150 \times \left( \frac {20}{100} \right) =$ ₹30.

And, then (Sale Price) = (Marked Price) – Discount = 150 – 30 = ₹120.

The sale price of an item marked at ₹150, offered at a discount of 20% will cost ₹120 to a customer.

You can also calculate the discounted price using the formula: (Sale Price) = (Marked Price) $\times$ (1 – Discount Percent).

Sale Price = $150 \times \left(1 – \frac {20}{100} \right) = 150 \times \left(1 – 0.2 \right) = 150 \times 0.80 =$ ₹120.

## What is a Successive Discount?

Sometimes shopkeepers announce more than one discount on some of their items to increase the sale of that item. When two or more consecutive discounts are allowed one after another on the list price of an item, they are known as successive discounts.

When calculating successive discounts, the first discount is calculated on the list price of the item while the second discount is calculated on the discounted price after the first discount.

A single discount of 50% is not the same as successive discounts of 25% and 25% or successive discounts of 20% and 30%. Even successive discounts of 20% and 30% are not the same as successive discounts of 25% and 25%.

Let’s consider the situation considered in the beginning to understand the concept of successive discounts.

But first offering a discount of 45% and the other at successive discounts of 20%, 20%, and 10%.

To begin, let’s consider the marked price as ₹100

**Case 1:** Discount = 45% => Discount amount = 45% of ₹100 = $\left( \frac{45}{100} \right) \times 100$ = ₹45

So, Sale Price = 100 – 45 = ₹55

**Case 2: **First discount = 20% of ₹100 = $\left( \frac{20}{100} \right) \times 100 =$ ₹20

Price after first discount = 100 – 20 = ₹80

Second discount = 20% of ₹80 = $ \left( \frac{20}{100} \right) \times 80 =$ ₹16

Price after second discount = 80 – 16 = ₹64

Third discount = 10% of ₹64 = $ \left( \frac{10}{100} \right) \times 64 =$ ₹6.40

Price after third discount = 64 – 6.40 = ₹57.60

You can see from these calculations that in the case of a discount of 45%, the customer pays ₹55 for an item, whereas in case of successive discounts of 20%, 20%, and 10%, the customer pays ₹57.60 for an item marked at the same price.

## Successive Discount Formula – Successive Discount Trick

If the successive discounts d1, d2, and d3 are given on an item, then the sale price of that item is calculated by, $ \text{SP} = \left(1 – \frac{d_1}{100} \right) \times \left(1 – \frac{d_2}{100} \right) \times \left(1 – \frac{d_3}{100} \right) \times \text{MP}$, where SP is selling price and MP is marked price.

Let’s use the formula to calculate the sale price considered in the example above.

MP = ₹100, $d_1 = 20\%$, $d_2 = 20\%$ and $d_3 = 10\%$

$\text{SP} = \left(1 – \frac{20}{100} \right) \times \left(1 – \frac{20}{100} \right) \times \left(1 – \frac{10}{100} \right) \times100 = \left(1 – 0.2 \right) \times \left(1 – 0.2\right) \times \left(1 – 0.10 \right) \times 100$

$= 0.8 \times 0.8 \times 0.9 \times 100 =$ ₹ $57.60$

## Let’s Code With Python

Find the effective discount when the successive discounts are $10\%$, $20\%$, and $30\%$.

#Ask for input d1 = float(input('Enter first discount ')) d2 = float(input('Enter second discount ')) #Calculate effective discount d = (1 - (1 - d1/100)*(1 - d2/100)) print('Effective discount is ', "{:.2f}".format(d*100), '%') # {:.2f}".format(d*100) prints with 2 decimal places.

**The above code calculates an effective discount when two successive discounts are given. Try modifying the code to calculate the effective discount, for $n$ successive discounts.**

## Practice Problems

- Three successive discounts of $5\%$, $10\%$, and $15\%$ are given. What will be the net discount (in percentage)? Will there be a change in net discount, if the order is changed as
- $10\%$, $15\%$ and $5\%$
- $15\%$, $5\%$ and $10\%$

- The marked price of a table is ₹$2,000$. The successive discounts offered by the store are $15\%$ and $20\%$. Calculate the total discount and the selling price of the table.
- The list price of a cupboard is ₹$8,000$. If the successive discounts offered by the store owner are $20\%$ and $22\%$. Then calculate the total discount and selling price of the cupboard.
- Suppose an additional discount of $25\%$ is offered on the cupboard in the previous question. Then, calculate the total discount and new selling price of the cupboard.
- Three successive discounts of $10\%$, $x\%$, and $10\%$ are given such that the effective discount becomes $35.20\%$. What is the value of $x$?
- The cost price of a book is ₹$150$. If the successive discounts offered on the book are $5\%$, $10\%$, and $15\%$. Then, calculate the total discount and selling price of the book.
- The marked price of a shirt is ₹$1,850$. If successive discounts offered by the store are $10\%$ and $x\%$. If the selling price of the shirt is ₹$1,400$. Then, calculate the value of $x$ and the total discount offered.
- The list price of a t-shirt is ₹$600$. If successive discounts of $15\%$ and $x\%$ are applied on the t-shirt and the selling price is found to be ₹$450$. Then, find the value of $x$.

## FAQs

### which is better than a successive discount of 40 and 30 or a discount of 70%?

Let the MP = 100.

In case of a successive discount of 40 and 30, $SP = \left(1 – \frac{40}{100} \right) \times \left(1 – \frac{30}{100} \right) = \left(1 – 0.4 \right) \times \left(1 – 0.3\right) \times 100 $

$= 0.6 \times 0.7 \times 100 = 0.42 \times 100 = 42$

In case of a single discount of 70%, S.P. = 100 – 70 = 30.

Therefore, a single discount of 70% is better than a successive discount of 40% and 30% for a customer.

### What is the formula of 2 successive discount?

If the successive discounts $d_1$, and $d_2$ are given on an item, then the selling price of that item is calculated by, $\text{SP} = \left(1 – \frac{d_1}{100} \right) \times \left(1 – \frac{d_2}{100} \right) \times \text{MP}$, where SP is selling price and MP is marked price.

## Conclusion

A successive discount is basically a discount on a product multiple times i.e. one discount after another. It seems to a customer that successive discounts are better than one single discount. But the fact can be the other way round. It’s better to calculate and compare the prices before making any decision.