Business Mathematics: Buying - Chain Discounts

As an extra incentive to purchase a product, some companies and retail stores offer chain discounts. To the uninformed consumer such discounts give the illusion of being larger than they really are.

Example
A stereo unit has a list price of $1,000. It goes on sale at a discount of 30% and then goes on clearance for an additional 20% off. What is the final price of the stereo?

The actual selling price is $560.
At first glance, it might appear that the combined discount should be 30% + 20% = 50%. If that were true, however, the price of the $1,000 stereo would be $500. The discount in example 7 was somewhat less than 50%.

It is a bit more convenient to use complements to compute chain discounts. Since
Price after 1st discount = (100% – 1st discount rate) × list price
and
Price after 2d discount = (100% – 2d discount rate) × price after 1st discount
it follows that
Price after 2d discount = (100% – 2d discount rate) × (100% – 1st discount rate) × list price
The order in which the discounts are taken is unimportant; the final selling price is the same if you reverse the order of the discounts.

Example
Consider the stereo in Example 7. Compute the final selling price using complements:

Method 1:
Price after 1st discount = (100% – 30%) × $1000 = 0.70 × $1000 = $700
Price after 2d discount = (100% – 20%) × $700 = 0.80 × $ 700 = $560 (final price)

Method 2:

Method 2 is a much more direct way of solving the problem of chain discounts. It extends in a natural way to three or more successive discounts.

Example
An evening gown lists for $1,500. It is offered at a discount store for 30% off list price. Since it does not sell, the store offers a second discount of 40% off the discounted price. It still does not sell. In desperation, the store offers 50% off the lowest sale price. What is the final price of the gown?

To make intelligent comparisons, it is useful to compute the single discount which is equivalent to a chain of successive discounts. Such a number is known as the single equivalent discount.

Example
A $400 barbecue is put on sale at 30% off. At the end of the summer, it is put on clearance for 50% off the sale price. What is the single discount equivalent to the chain discounts of 30% and 50% off?

The final price of the barbecue is $140, which is $260 off the original price of $400.

Even though the actual discount taken was broken up into two successive discounts, the problem can be viewed as one discount of $260.

As such, the

In other words, the chain discounts of 30% and 50% are equivalent to the single 65% discount.

To compute the single equivalent discount, follow these steps:
(a)  Compute the complements of each individual discount.
(b)  Multiply the complements and express as a percent.
(c)  Subtract from 100%.

Example
Use these steps to compute the single equivalent discount in Example 10:
(a)  The discounts are 30 and 50%. Their complements are 70 and 50%, respectively.
(b)  70% × 50% = 0.70 × 0.50 = 0.35 = 35%
(c)  100%–35% =65%
The single equivalent discount is 65%.

Solved Problems:

8.7 What is the selling price of a cashmere coat which lists for $400, goes on sale for 40% off, and then goes on clearance for an additional 20% off the sale price?

Solution A


Solution B


Solution C


8.8 The list price of a dining-room set is $3,500 with discounts of 20% and 30%. What is the final selling price?
 

Solution


8.9 Compute the sale price of a kitchen stove whose list price is $1,400 if it is given successive discounts of 25% and 40%. Verify that the price is the same regardless of the order in which the discounts are taken.
 

Solution
25% discount followed by 40% discount:

40% discount followed by 25% discount:

Either way the price after both discounts is $630.

8.10   Determine the single discount equivalent to a chain discount of 25% and 40%. Then compute the price of the stove in problem 8.9.
 

Solution
The complements of the two discounts are 75% and 60%, respectively:

The single equivalent discount is 55%.
Price of stove = (100% – 55%) × $1,400 = 45% × 1,400 = 0.45 × 1,400 = $630

8.11   Which is the best deal?
(a)  A 10% discount followed by a 60% discount
(b)  A 20% discount followed by a 50% discount
(c)  A 30% discount followed by a 40% discount
 

Solution
We compute the equivalent single discount for each case:
(a)  (100% – 10%) × (100% – 60%) = 0.90 × 0.40 = 0.36 = 36%:
Single equivalent discount = 100% – 36% = 64%
(b)  (100% – 20%)× (100% – 50%) = 0.80 × 0.50 = 0.40 = 40%:
Single equivalent discount = 100% – 40% = 60%
(c)  (100% – 30%) × (100% – 40%) = 0.70 × 0.60 = 0.42 = 42%:
Single equivalent discount = 100% – 42% = 58%
Clearly (a) gives the largest discount.

8.12   Compute the single equivalent discount of three successive 20% discounts.
 

Solution