3.7 SAT Math - In Terms of Strategy (ITOS)

What do you do if many of these problems seem far more confusing than you feel ready to handle?  Or what if you’re a slow test-taker, worried about the time figuring all these out algebraically will take?  What if you just don’t want to do all that work?  Don’t worry!  A fantastic SAT Math Strategy is here!

In Terms Of Strategy, or ITOS. 

To understand the In Terms Of Strategy, you first have to know what a variable is.  It represents an unknown, correct?  Can that unknown be any number?  Sure.  That’s why we use a variable – to represent some unknown until we can figure it out, or to write an equation that will be true for all unknowns.  Sometimes that variable represents a very specific number, like in the equation 4x + 2 = 10. You have to do a little work to figure it out, but that x most definitely represents the number 2.  Sometimes, however, that variable represents a whole lot of different numbers, like if a question says, “Paul is a waiter.  He has figured out that if he works h hours, he makes 2.15h + .2s - (.08(.2sh) where s represents his total sales for the day.”  In this case, Paul’s hours worked and Paul’s sales for the day will vary every single time he does this equation, but the equation itself should still hold true despite those changing numbers.  If you are presented with that type of question, there is a really fantastic technique you can use to get you out of doing a ton of work that you might not even know how to do, and at the very least will take you a lot of time.  This SAT Math Strategy is called ITOS, or In Terms Of Strategy.

How do you know when you can use it?  When at least one answer contains a variable (not equations, variables).  Often, the words “in terms of” will also appear in the question, but they don’t have to.  So how does it work?

1) Recognize a problem in which you can use it.  For which of the two following examples can ITOS be used?

2. For any positive odd consecutive sequence beginning with integer x, which of the following will be one of the following numbers in the sequence?
(A) x + 1
(B) + x
(C) 0
(D) x + 1
(E) x + 4

4. If 2a + 2b = 14, then a + b =
(A) 2
(B) 7/2
(C) 7
(D) 10
(E) 28

The Number 2 question above contains variables in the answer choices; the Number 4 does not.  It doesn’t matter that one of the answer choices for Number 2 is a number – if even one of the answer choices contains a variable, you can use ITOS.

2) Substitute numbers for all variables in the problem.

Using example Number 2 above, pick a number to substitute in for x that satisfies the limits of the question.  What are those limits?  It must be positive, it must be odd, and it must be an integer.  Let’s try 7.

3) Using your numbers, figure out the answer to the question.  Put a circle around that answer so that you can easily find it.

So using 7 for x, what are some numbers in a consecutive odd sequence? 7, 9, 11, 13, 15…  So now, let’s look to the answer choices.  Using 7 for x, plug that number into each answer choice to see if you get a match.
(A) 7 - 1 = 7   NO – the question said it must be odd
(B) -7              NO –the question said the sequence must start with 7 and be positive
(C) 0                NO – the question said the sequence must start with 7 and be odd
(D) 7 + 1 = 8   NO – the question said it must be odd
(E) 7 + 4 = 11   YES  This is one of the possible answers we found, so this is correct.

 Let’s try one a little harder.

Example Problem 1

19.  If x and y are positive integers, then 8x•2(x + y) =
(A) 24x + y
(B) 23x2 + 3xy
(C) 16x2 + y
(D) 16x2 + 16y
(E) 16x(x + y)

This is a very difficult question that under normal circumstances, most students with a starting score below 650 or 700 should not even attempt.  But let’s try it using ITOS.

1) Can we use ITOS?  Sure.  There are variables in the answer choices.

2) Substitute numbers for each variable.  Let’s try 3 for x and 5 for y.

3) Using your numbers, figure out a numerical answer to the question.
Plug 8³•2^{(3 + 5)} into your calculator.  What do you get?

131072, right?  (Hint:  If you didn’t get this, check your work.  Did you multiply when you were supposed to? Add?  Use parentheses?)

Circle this number.  Now, let’s go to the answer choices.
(A) 2^4•3+5 = 131072   MATCH!  But we’re not done yet.  We have to check to make sure it’s the only one.
(B) 2^{3•32 + 3•3•5}  = 4.72 x 10²¹ NO    (If your calculator reads 4.72... ∈ 21, this is the same thing.  It’s how many calculators show scientific notation.)
(C) 16^{32 + 5} = 7.21 × 10¹⁶    NO
(D) 16³² + 16⁵ = 68720525312 NO
(E) 16³⁽³⁺⁵⁾   = 7.92 x 10²⁸ NO

The answer is (A).  Entering these into your calculator didn’t take very long, and now you have an answer to a Number 19 question, without much work.

Sometimes you have to choose a number for one variable and then solve to find the other, but the Strategy still works the same.

Example Problem 2

16. If x = (w - 3)2 then (-2w + 6)2 must equal which of the following?
(A) +4x2
(B) +2x2
(C) 2x
(D) 4x
(E) x2

Which variable should we start with?  Always start with the messier one, the one with more stuff around it.  So in this case we’ll start with w.  Let’s try w = 7.  To figure out what x must equal, we plug in our w and see what we get. (7 - 3)² = 16   so x = 16.  Using w = 7, figure out the next part. (-2•7 + 6)² = 64   So which number do you circle before checking the answer choices?   Reread what the question is asking for – it’s looking for what number that second part finds, so 64 is the number you circle and try to match with the answer choices.
So now, go to the answer choices, using x = 16, and see which one gives you 64:
(A) -4•16² = -1024
(B) -2•16² = 512
(C) 2•16 = 32
(D) 4•16 = 64
(E) 16² = 256

So (D) is our answer choice.

You may have noticed that we keep picking numbers like 3, 5, and 7.  ITOS will work with any numbers but some numbers tend to work too well, producing two answers that both seem right.  Let’s try that last example again, this time using w = 3.  So for the first part we get (3 - 3)² = 0, which means x = 0.  For the second part we get (-2•3 + 6)² = 0.  See what happened?  We got 0 for everything, which means all the answer choices are going to work out to 0 as well.  Totally not helpful.
What if we had tried w = 1?  We would get x = 4 for the first part, and work the second part out to be the number 16.  Going to the answers with x = 4, we would get:
(A) -4•4² = -64
(B) -2•4² = -32
(C) 2•4 = 8
(D) 4•4 = 16
(E) 4² = 16

What happened here?  We got 16, the right answer, for two different choices.  If either of these two scenarios happens, don’t panic.  Simply pick new numbers, do the work with the new numbers, and then JUST check the two that gave you the same answer to see which one is a coincidence, and which one is correct.

The best way to prevent this from happening is to avoid bad numbers. 

• Never throw in 0 or 1 for your variables, or 2 if you have exponents. 
• Avoid numbers that are going to cancel each other out, so if you are subtracting 3 in the problem, don’t choose 3.  If you’re dividing by 4, don’t choose 4. 
• If you have multiple variables to choose numbers for, pick different numbers for each variable to prevent them canceling each other out. 
• Pick numbers that are easy to work with in the question, even if they’re unrealistic.

Let’s try another:

Example Problem 3

18. The price of corn meal is d dollars for 12 ounces and each ounce makes c squares of cornbread.  In terms of c and d, what is the dollar cost of the corn meal required to make 1 square of cornbread?
(A) dc/12
(B) 12/dc
(C) d/12c
(D) c/12d
(E) 12dc

Pick good numbers here.  It may be more realistic to use $1.99 for 12 ounces of corn meal, but that’s going to be extremely annoying to work with.  Use something easy, like d = 24, so that you get an easy to work with answer full of as many whole numbers as possible.

Keep your work neat.  The only downside of ITOS is that you’re not really doing the real work your math teacher would want you to do, so it’s hard to see if you’ve made a careless mistake.  The only insurance you have is that neat, written work.

(C) is the answer.

That’s it for ITOS!  Use it as often as you recognize it to help you shortcut through difficult problems.