haidgmat wrote:Can someone please explain this to me?
Henry purchased 3 items during a sale. He received a 20% discount off the regular price of the most expensive item and a 10% discount off the regular price of each of the other 2 items. Was the total amount of the 3 discounts greater than 15% of the sum of the regular prices of the 3 items?
1. The regular price of the most expensive item was $50, and the regular price of the next most expensive item was $20
2. The regular price of the least expensive item was $15
If you have a solution other than the one in the book that would GREAT!
The answer is A, not C. Conceptually, this is just a weighted average problem, though that fact is a bit disguised. The idea is this: suppose you buy two items, X and Y. On item X you get a 20% discount, and on item Y you get a 10% discount. Your overall discount is thus somewhere between 10% and 20%:
* if X and Y cost the same amount, your overall discount will be 15%
* If X costs more than Y, your overall discount will be closer to 20%
* If Y costs more than X, your overall discount will be closer to 10%
So here, think of the two cheaper items as a single purchase. From Statement 1, we know that the two cheaper items cost at most $40, so they cost less than the most expensive item, which cost $50. Thus our overall discount must be greater than 15%. Statement 2 is clearly insufficient, so the answer is A.
You can also prove all of this algebraically. From Statement 1 we know our expensive item costs $50. Say our two other items cost $x in total. Then our total discount, in dollars, is (0.2)(50) + 0.1x = 10 + 0.1x. To find our percent discount, we divide this by the total expenditure, which is 50+x (here I'll write each percentage as a decimal, so I won't multiply by 100, and we'll write 15% as 0.15). If we want to know for which values of x this overall discount will be greater than 15%, we can set up the following inequality:
(10 + 0.1x) / (50 + x) > 0.15
10 + 0.1x > 7.5 + 0.15x
2.5 > 0.05x
50 > x
So provided the two less expensive items cost less than $50 in total, the overall discount will be greater than 15%. Since Statement 1 guarantees that the two less expensive items cost less than $40, it is sufficient.
The concept in the question above is one you encounter more often in mixtures questions. If you combine, say, 50 Liters of a 20% salt solution with 50 Liters of a 10% salt solution, you'll get a solution which is exactly 15% salt. If you combine 50 Liters of the 20% solution with, say, 40 Liters (or any other amount less than 50 Liters) of the 10% solution, the resulting concentration will be greater than 15%.
