He saves$10 on each chair, and $12 on each table. So using Statement 1, we know
10c + 12t = 100
Dividing by 2,
5c + 6t = 50
If you have an equation where you are adding or subtracting two integers on one side, and getting some result on the other side, and two of the terms share some common divisor, then the third term must also be divisible by that divisor. Here two of our terms are divisible by 5, so the third term must be as well. That is, 6t must be divisible by 5, so t must be divisible by 5. Since t cannot be 10 or more (then c would need to be negative), t must be 5, and the equation has only one solution. t cannot be 0, because the question tells us c and t are positive.
If it's not clear why t must be divisible by 5, you can rearrange the equation with the multiples of 5 on one side:
6t = 50 - 5c
6t = 5 * (10 - c)
and now, since 6t is the same number as 5 * (10 - c), it must be true that 6t has the same divisors as 5 * (10 - c), and since 5 is clearly a divisor of 5 * (10 - c), it must also be a divisor of 6t.
Statement 2 tests weighted average principles. If you get different percent discounts on two items, your overall percent discount is a weighted average of the two percent discounts, weighted by the original prices of the two items. If you know the 'alligation' method (or 'number line method') for weighted average problems, and if you recognize this as a weighted average problem, it will be immediately clear we can use Statement 2 to find the ratio of the original cost of the chairs to the original cost of the tables, and from there find the ratio of c to t and answer the question. If you are not familiar with that method, we can also use algebra. The amount he spent, after discount, was 40c + 68t. The amount he would have spent, with no discounts, was 50c + 80t. We know the first number is 5/6 of the second:
40c + 68t = (5/6) (50c + 80t)
20c + 34t = (5/6) (25c + 40t)
120c + 204t = 125c + 200t
4t = 5c
t = (5/4)c
So we can find the ratio of t to c, and with that information we can answer the question, since the question asks us to find 40c / (40c + 68t), and if we substitute now for t, the letters will all cancel out.
It could be realistic to see either one of these two statements in a real GMAT question, but not both in the same question, since for most test takers it would take too long to analyze each of these statements individually.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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