As mentioned above, we encounter mathematical challenges in our everyday lives and don't even realize that they can be properly understood with just a little mathematics know-how. We often visit stores that are running a sale and then on a special day will add a percentage on top of the one that was previously advertised. The typical response is to add the two percentages and conclude that the total savings for the day would be the sum of the two percentages. Upon close reflection, you will see that this is clearly a wrong calculation. Most folks defer thinking about percentage problems, as they see them as nothing but a nemesis. Problems get particularly unpleasant when multiple percentages need to be processed in the same problem. However, we shall see how such successive percentages lend themselves very nicely to a delightfully simple arithmetic algorithm that leads us to lots of useful applications and provides new insight into successive percentage problems. We think you'll find this not-very-well-known procedure enchanting.
Let's begin by considering the following problem:
The problem: Wanting to buy a coat, Barbara is faced with a dilemma. Two competing stores next to each other carry the same brand of coat with the same list price, but with two different discount offers. Store A offers a 10 percent discount year-round on all its goods, but, on this particular day, it is offering an additional 20 percent discount on top of its already-discounted price. Store B simply offers a discount of 30 percent on that day in order to stay competitive. Are the two end prices the same? If not, which gives Barbara the better price?
At first glance, you may assume there is no difference in price, since 10 + 20 = 30, which would appear to be yielding the same discount in both cases. Yet, with a little more thought, you may realize that this is not correct, since in store A only 10 percent is calculated on the original list price, while the 20 percent discount is calculated on the lower price (that is, the 10 percent discounted price); in comparison, at store B, the entire 30 percent is calculated on the original price. Now, the question to be answered is, What percentage difference is there between the discount in store A and store B?
To determine the difference in the prices, one procedure might be to assume the cost of the coat to be $100, and then calculate the 10 percent discount, yielding a $90 price, and then calculate an additional 20 percent of the $90 price (or $18), which will bring the price down to $72. In store B, the 30 percent discount on $100 would bring the price down to $70, giving a discount difference of $2 between the two stores, which in this case would be a 2 percent difference. This procedure, although correct and not too difficult, is a bit cumbersome and does not always allow a full insight into the situation, as you will soon see.
We shall provide an interesting and quite unusual procedure for a deeper look at this situation as well as for entertainment.
We will consider a somewhat mechanical method for obtaining a single percentage discount (or increase) equivalent to two (or more) successive discounts (or increases). Follow this four-step procedure:
(1) Change each of the percentages involved into decimal form:
0.20 and 0.10
(2) Subtract each of these decimals from 1.00 if you are calculating a discount or decrease (for an increase, add to 1.00):
0.80 and 0.90
(3) Multiply these decimals:
(0.80)(0.90) = 0.72
(4) Subtract this number from 1.00:
1.00 – 0.72 = 0.28, which, written as a percent, is 28 percent. This represents the combined discount.
(If the result of step 3 is greater than 1.00, subtract 1.00 from it to obtain the percent of increase.)
When we convert .28 back to percent form, we obtain 28 percent, which is the equivalent of successive discounts of 20 percent and 10 percent.
Therefore, we can conclude that the combined percentage of 28 percent differs from the single discount of 30 percent by 2 percent. As such, in our example above, Barbara should purchase her coat from Store B.
Following the same procedure as above, you can also combine more than two successive discounts. Furthermore, successive increases, combined or not combined with a discount, can also be accommodated in this procedure by adding the decimal equivalent of the increase to 1.00 (recall, the discount was subtracted from 1.00), and then continue with the procedure in the same way. If the end result comes out greater than 1.00, then this will have resulted in an overall increase rather than the discount as found in the above problem.
A conundrum often facing consumers is that of determining whether a discount and increase of the same percentage leaves the original price unchanged. For example, suppose a store just increased all of its prices by 10 percent and then notices that its business has declined substantially, whereupon they then resort discounting all of these recently increased prices by the same percentage: 10 percent. Have they then restored the prices to their original level? Using this technique, we find ourselves multiply 1.1 times 0.90 to get 0.99, which would indicate that the original price had dropped by 1 percent. For many people this is a counterintuitive result. (Again, with further reflection, you'll see that the difference is a result of considering the original price prior to the 10 percent increase and then the new price prior to its 10 percent decrease. Ten percent of the lower price amounts to less than 10 percent of the higher price. By taking 10 percent off of the higher price, the store is therefore providing a steeper discount than it had for the increase.)
As you can see, this procedure not only streamlines a typically cumbersome situation but also provides some insight into the overall picture. For example, consider the following question: Is it advantageous to the buyer in the above problem, Barbara, to receive a 20 percent discount and then a 10 percent discount, or the reverse, a 10 percent discount and then a 20 percent discount? The answer to this question is not immediately intuitively obvious. Yet, since the procedure just presented shows that the calculation is merely multiplication, a commutative operation, we can immediately conclude that there is no difference between the two.
So here you have a delightful algorithm for combining successive discounts or increases or combinations of these to calculate the combined result. Not only is it useful, but also it gives you some newfound power in dealing with percentages when a calculator might not be available.
Another shopping situation in which mathematics can be helpful is when there are discounts of different types. Suppose you have two sales-promotion coupons for the same store, one that says “20 percent off” (independent of the purchase) and one that applies only if a certain minimal amount of money is spent, for example, “$15 off for purchases exceeding $49.99.” Assuming that the two coupons cannot be combined, then which one would be more advantageous to use, if the item we want to buy costs, say, $80? The 20 percent off coupon would yield a price of $64, while the $15 reduction would yield a price of $65. It might be nice to know at which price the coupon for 20 percent off will become the more advantageous option.
To approach this problem, we may consider two extreme cases. Let's consider an item costing $50, since the minimal purchase that qualifies for the second coupon is $50, which when reduced by $15 would be $35. On the other hand, the fixed-percentage coupon would yield a reduction of 20 percent, or one fifth, of $50, which is $10. That means we would have to pay $40, if we used the fixed-percentage coupon. Thus, the $15-off coupon would be the better choice for a $50 purchase.
The other extreme case would be infinity, but since we can spend only a finite amount of money, let's assume our budget is very high, say it is $150. One fifth of $150 is $30. Therefore, the 20 percent coupon yields a reduced price of $120, while the other coupon (a reduction of $15) would result in a price of $135. Obviously, the better choice of coupon depends on the total sum of the purchase. For the fixed-percentage coupon, the amount of money saved increases with the price, whereas we cannot save more than $15 with the other coupon. The extreme cases we have considered show that there must be some break-even point X between $50 and $150 at which price both coupons yield the same discount. For purchases at $50, the 20 percent reduction will be $10 and the $15-off coupon is therefore preferred; but, for purchases at or above $50 and below X, the $15-off coupon is still the better choice. To find the break-even price X for a 20 percent coupon and a $15-off coupon, we just have to compute: X – 0.20X = X – $15, and then X = $75. Therefore, if we want to buy an item for more than $75, we should use the 20 percent coupon.
Occasionally you may also encounter different types of coupons that are combinable, although this is a rare phenomenon, since most stores are usually not that generous to their customers. Let's take a look at such a situation, since it provides an example for the mathematical notion of noncommutativity. Suppose we were allowed to use both coupons for the same purchase, that is, coupon A with a 20 percent discount as well as coupon B with a $15 reduction. Now the question arises whether or not the order of the discounts matters. If it were to matter, which of the two coupons should be used first? Denoting the price without any discount by P (which we assume to be at least $50 for the sake of simplicity), we obtain the following:
a reduced price by using coupon A first is pA,B = P · 0.8 – $15, and
a reduced price by using coupon B first is pB,A = (P – $15) · 0.8
To summarize, applying coupon B first gives us pB,A = (P – $15) · 0.8 = P · 0.8 – $12; because that is more than pA,B, which is P · 0.8 – $15, we should apply coupon A first, unless P · 0.8 is less than $50. In this case, the $15-off coupon (B) would not be applicable anymore and we would have to use coupon B first to get the more favorable price.
Two operations that will in general lead to different results if their order is reversed are called “noncommutative” in mathematics. As our analysis showed, these different types of discounts are an example of noncommutative operations, meaning that the order does matter! You should think about that if you are offered combinable discounts of different types. However, requesting your preferred order might be a bit of a challenge.