There's one more tool to add to your QC Game Plan. This tool is specialized, like a particular kind of wrench. You can only use it in certain circumstances. But when you can use it, it can open up the problem quickly and easily.
What are those specialized circumstances? The Quantity B column has to contain just a number. Meanwhile, Quantity A will be something more complicated, but you can compare it to Quantity B in a straightforward way. And there'll often be some upfront information, giving you the context and enough information to solve the problem.
For instance:
| Some information about a chicken and a road, including the chicken's constant speed and the distance across the road. |
|
| Quantity A | Quantity B |
| The number of seconds it takes the chicken to cross the road |
20 seconds |
In those circumstances, you can break out the special wrench. You can Pick Numbers—specifically, the number in Quantity B. That is, you can cheat off B.
For the problem here, you'd pretend that 20 seconds is the number of seconds it takes the chicken to cross the road. Then you'd work backward, using the information given up front. It should become apparent whether 20 seconds is too short of a time, too long of a time, or just right.
Again, when you pick the number in Quantity B, you're working backward. Take this problem, for example:
A discount of 30% off the original selling
price of a dress reduced the price to $99.
| Quantity A | Quantity B |
| The original selling price of the sweater |
$150 |
You can absolutely go "forward" through this problem. Name a variable for the original selling price of the dress, say P. Set up an equation to represent the upfront information: P − 0.3P = 99. Solve for P. Make sure that this "forward" method works for you.
But take a look at how you can cheat off B. Assume the original price was $150, the value of Quantity B.
If the original price was $150, and it was reduced 30%, you can use the calculator (or your mind) to figure out the discounted price very quickly: 30% of 150 is 0.3 × 150 = $45. That means the discounted price is $150 − $45 = $105.
An original price of $150 with a 30% discount would have made the new price $105. That is higher than the actual discounted price you were told ($99).
So the original price of the sweater must have been less than $150. The answer is (B).
Even without upfront information, Quantity B can help. Earlier in this chapter, we talked about trying to prove (D). Sometimes, with Quantity B as your guide, the best way to try to prove (D) is actually to try to prove (C). That is, assume that Quantity A equals Quantity B, and see what happens.
Have a go at this problem:
| Quantity A | Quantity B |
| The perimeter of Triangle ABC, an isosceles triangle whose longest side is equal to 11 |
22 |
Okay, can you imagine a triangle with a perimeter greater than 22? Sure. Make one of the other sides 11 too (there's the isosceles) and the third side 10. The perimeter will be greater than 22.
So now, trying to prove (D), you want to find a triangle that has a perimeter of 22 or less. Let's start with the equality: make Quantity A equal 22, and work backward.
The number 22 gives you a goal, so that you do not have to search blindly and create random isosceles triangles that get you no closer to an answer.
If one of the sides is 11, that means that the remaining two sides must have a combined length of 11 if you are to achieve a perimeter of 22. You have already seen what happens if the two equal sides each have a length of 11. So, for the triangle to remain isosceles, the two unknown sides must be equal. The only way they could be equal is if they each have a length of 5.5.
Careful! There is a trap here. Remember, any two sides of a triangle must add up to greater than the length of the other side, or else you can’t connect all three sides with space in the middle for the actual triangle. A "triangle" with sides 11, 5.5, and 5.5 is actually completely collapsed into a single line segment of length 11.
So this triangle does not in fact exist. You can't make the two other sides 5.5 units in length. And you can't make them even shorter, either.
Putting it all together, you know that the perimeter of Triangle ABC will be greater than 22. The answer is (A).
By specifically trying to make the two values equal, you were able to prove that Quantity A will always be greater. Cheating off B and trying to prove (C) broke open the problem.
Let's adjust the QC Game Plan one more time, adding this last technique.
| 1. Simplify | → | 2. Pick Numbers |
|
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The master plan is still just "Simplify, then Pick Numbers." Repeat that aloud to yourself. Walk into the exam with that mantra: Simplify, then Pick Numbers… Simplify, then Pick Numbers…
As for the bulleted tactics, you won't use every one of them on every QC problem. But you'll use at least a couple of them, maybe in succession. On some problem, maybe you'll first unpack some upfront information. Then you'll simplify across the Hidden Inequality. Finally, you'll pick a couple of numbers, trying to prove (D). With practice, you'll get good and fast at switching from tactic to tactic.
The remaining chapters in this unit explore specific QC examples by broad content area: Algebra, Fractions, Decimals, & Percents, Geometry, Number Properties, and Word Problems.