6.6 SAT Math - SAT Strategy: Solve the Relationship

A similar type of SAT question to Must/Always is Solve the Relationship.  This strategy is a combination of ITOS and Using the Answer Choices, and you can use it when a question is asking you for a relationship between two numbers rather than a specific answer.  Relationships are things like, "what is the ratio of a to b" or "what percent of the value" or "what could be the sum of the two numbers?"

These questions are similar to ITOS in that you want to throw in your own numbers, but instead of variables in the answer choices, there are numbers.  The similarity comes in because, like dependent variables in ITOS, you're going to need to put in a number for one of the variables, and use that number to solve for all of the other variables.  Using all those numbers, you're going to see which answer matches what you got.  The steps to follow to use this strategy are pretty simple:

1. Pick ONE variable and change it to a number of your choice.
2. Using that number, solve for all of the other variables in the question.
3. Using those numbers, do the math required by the question.
4. See which answer choice matches what you got with your numbers.

The cool thing is, when a SAT question is asking for a relationship, that relationship never changes even when the numbers do.  Think about ratios for a moment.  If the ratio of boys to girls in a class is 2:3, it doesn't matter whether there are 8 boys or 102.  The relationship will always be 2:3.  That's why Solve the Relationship works.

Example Problem

Take a look at this one:

18. The variables a, b, c, and d are all positive integers.  If a/b = c and c/d = b, what is b when a/d = 4?

(A) 1/4
(B) 1
(C) 2
(D) 4
(E) 8

Start by picking a number for one of the variables.  The easiest to start with will be a or d since those are the only two that we have any numerical information on, and the easiest of those is probably a since it's in the numerator.  Let's try 16 for a.  That gives us 4 for d.  Write in a neat list: a = 16 and d = 4 on your paper so you don't lose track.

Now, using those numbers, you need to solve for b and c.  Let's set up the next two equations, subbing in the numbers we know:

16/b = c and c/4 = b
Again, we could solve for either b or c here, but since c is in the numerator, it's probably easier.  Let's use 8.  Eight divided by 4 will give us 2, and now we have double-check with the other equation to confirm: 16/2 = 8.  Yup, it works.  So b equals 2 and the answer is C).

            -Hint: Pick easy to work with numbers!  If you decided that a was going to equal 6 before you even really looked at the question, and then started to work with 6/d = 4 to solve for the others….stop.  Go back.  Pick a better number for a that's going to divide evenly.  You don't have to stick with the first number you try if your instinct is wrong; just don't get frustrated, try a bunch, and lose track of what you're doing.  If you start getting messy fractions, try a new number, but then work with what you've got.  You want any strategy to make your life easier not more difficult.

Try this one on your own:

11. If Sam's age is twice the age Kelly was two years ago, Sam's age in four years will be how many times Kelly's age now?
(A) .5
(B) 1
(C) 1.5
(D) 2
(E) 4

[The answer is D) 2.]