Successive Discounts:



Introduction:

Normally, discount is the amount deducted from the marked price or tagged price of an article. When we deduct discount amount from the Marked Price or tagged price of any article, we get the selling price. But, when the discount is again deducted from the selling price and a new selling price is obtained, then this condition is called having successive discounts. Discount Rates in Successive Discounts can have different rates or same rates i.e. discount rates could be 20% and 20% or 20% and 30%.

Definition:

Discount given on the Selling Price of an article that has already beared a discount on the Marked Price is called successive discounts. 



Understanding:

For example, we have an article costing Rs 10000 which is its' Marked Price or Tagged Price. When we give discounts of 10% and 20%, successively on this article; the selling price we get after this, is said to be the Selling Price after Successive Discounts.

To be more clear, we will have two Marked Price (for understanding) in successive discounts.

Solving the above example, we would get:
Solution:
Given,
Marked Price of an article (M.P.)= Rs 10000
Successive Discounts = 10% and 20%

Now,
S.P. = MP - 10% of MP
= 10000 - 10% x 10000
= 10000 - 1000
= Rs 9000

Again, 
Required S.P. = Rs 9000 - 20% of RS 9000
= 9000 - 20% x 9000
= 9000 - 1800
= Rs 7200

So, this is how we solve these types of questions having successive discounts.


Generating Formulae:

But, can we get some Formulae to do the work a lot easier and faster? Instead of solving the question in two steps, can we just find the solution within one step? Okay, Let's find it out by generating the formulae.

First of all, we need to understand this:

      S.P. = M.P. - d% of M.P.
or, S.P. = M.P. (1 - d%)
or, S.P. = M.P. (1 -d/100) .... (i)

Above, we took Marked Price common in the exdivssion to bring the factors of the formulae.

Successive Discounts can be of two types: Same discount rates or different discount rates.

Let, M.P. = x
Discount 1 = y%
Discount 2 = z%

Formula for Same Discount Rates:

We have, 
S.P. = x (1- y%)
or, S.P. = x(1 - y/100) .... (a)

If we have same discount rates, Discount 2 = y%

Now, our M.P. is exdivssion generated in equation (a)

This means:
M.P. = x(1 -y/100)
Discount = y% of M.P.
Now,
S.P. = M.P. - d% of M.P.

= x (1- y/100) - {y% of x(1-y/100)}
= x (1- y/100) - {y/100 * x(1-y/100)}
= x (1- y/100) (1 - y/100)
= x (1- y/100)^2

So, when there is successive discount of same rates then, we have the following formula:

S.P. = M.P. (1 - Discount /100)^2

Let us say that, we have three discount rates. Now for this, we already have the formula for two discount rates which will be our MP now;

MP =  x (1- y/100)^2
Discount = y % of MP

So, SP = MP - y% of MP

=  x (1- y/100)^2 - {d% of  x (1- y/100)^2}
=  x (1- y/100)^2 ( 1 - y/100)
=  x (1- y/100)^3

So, 
the formulae of successive discounts when the discount rates are same goes in the sense:
 MP (1- Discount%/100)^n
where 'n' refers to the number of discounts.


Formula for Different Discount Rates:

We have;
M.P. = m
Discount 1 = x%
Discount 2 = y%
Discount 3 = z%

Normally, when we have only Discount 1, our SP will be:

SP = MP - d% of MP
or, SP = m - x% of m
or, SP = m (1 - x%)  ...... (ii)
So, SP = m (1 - x/100)

In successive discount rates having two discount, our MP is the SP we got in equation (ii);
So,
MP = m( 1-x%)
Discount 2 = y%
Now, 
SP = MP - d% of MP
or, SP = m(1-x%) - { y% * m(1-x%)}
or, SP = m(1-x%) (1 - y/%) ..... (iii)
So, SP = m(1-x/100)(1-y/100)

Formula for Two Successive Discount of Different Rates = 
 MP (1-x/100)(1-y/100)

Again, if there are three successive discounts, our MP becomes the SP when we had two successive discounts in equation (iii)
This means,
MP = m(1-x%)(1-y%)
Discount 3 = z%

Now,
SP = MP - d% of MP
or, SP = m(1-x%)(1-y%) - { z% of m(1-x%)(1-y%) }
or, SP = m(1-x%)(1-y%) ( 1 - z%)
So, SP = m(1-x/100)(1-y/100)(1-z/100)

Therefore, from above, we can conclude that;
the formula when there are two or three or four or more successive discounts with different discount rates, our formula to find them is:

MP(1- D1/100)(1-D2/100)(1-D3/100)...

 
In conclusion, successive discounts means having a discount on marked price first and then having more discount on the obtained selling price.

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