| x > 0 | |
| Quantity A | Quantity B |
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Let's revisit this problem from the last section, after making things match in the bases.
| Quantity A | Quantity B |
| 620 = (2 × 3)20 = 220 × 320 |
(910)(85) = (32)10 × (23)5 = 215 × 320 |
We noted that the 320 is the same on both sides, and then we ignored it. How did we know we could do that? And can we extend that idea?
Well, the essence of a QC question is "which side is bigger?" You're given two quantities, and you're trying to figure out what symbol goes between them:
| Quantity A | Quantity B | ||
| (A) | This | > | That |
| (B) | This | < | That |
| (C) | This | = | That |
| (D) | This | can't tell | That |
The way you start every QC problem is with a question mark in the middle, a question mark you're trying to figure out:
| Quantity A | Quantity B | |
| This | ? | That |
You can think of that question mark as a Hidden Inequality. (It might be an equals sign, too, of course.)
So you can simplify both sides of the Hidden Inequality at the same time, using algebraic moves you're allowed to make to both sides of an inequality without changing the inequality. For instance, you can add or subtract anything from both sides. Or you can multiply or divide by anything positive.
What we were really doing in the problem above was dividing both sides by 320, to get rid of it. That's a legal move, because you can divide both sides of an inequality by a positive number without a care in the world.
| Quantity A | Quantity B | |
| 220 × 320 | ? | 215 × 320 |
| ÷ 320 |
÷ 320 |
|
| = 220 | ? | = 215 |
That's why you can just compare 220 and 215.
This move is an extension of the Simplify step. Up to now, you've been simplifying within each column separately. But now you know that you can simplify across the columns too, as long as you respect the Hidden Inequality. Go ahead and do your algebra, both within and across.
Try another example:
| x > 0 | |
| Quantity A | Quantity B |
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You could try plugging in a number for x, but it would be time consuming, even if you pick a simple number like 1. Additionally, how would plugging in a single number for x convince you that the conclusion was always valid? You would still need to try to prove (D).
Why not do some algebra across the Hidden Inequality?
You're told that x > 0. So you’re allowed to multiply or divide both sides by x, which is a positive number. In fact, on further thought, you’ll be better off multiplying both quantities by 2x. That way, you'll get rid of the denominator in Quantity A.
Put a "?" in between the two quantities and multiply through by 2x.
The two quantities are equal. The answer is (C).
It's a good thing to be able to pick numbers on QC. But in some cases, algebra is the right move. Look for ways to simplify within a column or especially across the columns, across the Hidden Inequality.
Let's add these points to the QC Game Plan.
| 1. Simplify | → | 2. Pick Numbers |
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