Percentages Applied Successively ...

Came across a nice simple problem but an alternative insightful solution to it as well. I quite liked it. So posting it here.

Wanting to buy a coat, Lisa is faced with a dilemma. Two competing stores next to each other carry the same brand coat with the same list price, but with two different discount offers. Store A offers a 10% discount year round on all its goods, but on this particular day offers an additional 20% on topof its already discounted price. Store B simply offers a discount of 30% on that day in order to stay competitive. How many percentage points difference is there between the two options open to Lisa? Which one should she go for?

Discussion

Assume the cost of the coat to be 100,calculate the 10% discount,yielding 90 as the price,and an additional 20% on 90 (or 18) will bring the price down to 72. In store B, the 30% discount on 100 would bring the price down to 70,giving a discount difference of 2,which will be 2%.

Correct procedure. But came across another insightful way :

Here is a mechanical method for obtaining a single percentage discount equivalent to two (or more) successive discounts.

1. Change each of the percentages involved into decimal form: .20 and .10.
2. Subtract each of these decimals from 1.00: .80 and .90
3. Multiply these differences: .8 * .9 = .72

4. Subtract back from 1.00: 1.00-.72 = .28

5. Convert it back to percentage i.e 28%

So the store A's successive discounts of 20% and 10% is actually a cumulative discount of 28%. The store B's discount is 30%. The difference again as proven earlier is 2%.

This procedure not only streamlines a typically cumbersome situation,  but also provides some insight into the overall picture. For example,the question "Is it advantageous to the buyer in the above problem to receive a 20% discount and then a 10% discount,or the reverse,a 10% discount and then a 20% discount?". Since the procedure shown above clearly indicates that the calculation is merely multiplication, a commutative operation, we can immediately tell that there is no difference between the two.