For some of the more difficult, abstract questions on the ACT Math test, the “real” math that the test takers have in mind can be quite tricky. But for some of these problems, there might be a way to make the rather abstract problem more concrete, and thus easier. That leads us to an important ACT Math strategy: Picking Numbers.
Consider the following difficult problem which has previously appeared on the ACT.
For every positive two-digit number, x, with tens digit t and units digit u, let y be the two-digit number formed by reversing the digits of x. Which of the following is equivalent to x – y?
A. 9(t – u)
B. 9(u – t)
C. 9t – u
D. 9u – t
This is a very difficult problem if you deal with it abstractly. Instead, let’s pick some numbers for t and u. Let’s suppose we make t = 7 and u = 2. Then x, which has a tens digit of t and a units digit of u, is 72. We also know that y is the number you get when you switch the digits, so y is 27.
The question asks us to find the value of x – y. Using our numbers x – y = 72 – 27 = 45.
All we need to do now is plug our values for t and u into the answer choices and see which one of them is equal to 45.
A. 9(t – u) = 9(7 – 2) = 45 YES!
B. 9(u – t) = 9(2 – 7) = -45 NO
C. 9t – u = 9(7) – 2 = 61 NO
D. 9u – t = 9(2) – 7 = 11 NO
E. 0 NO
So the correct answer is Choice A!
Why did I keep testing the answer choices once I know that Choice A worked? This is actually very important to do. If you have bad luck, you might pick numbers that make two (or more) different choices appear to be true. Try this problem again with t = 9 and u = 5. See what happens? Both Choice A and Choice D appear to work! But there can’t be two correct answers. When that happens, you should choose new numbers and try the problem again.
Here’s a problem that you can try:
After the first term, each term in a sequence is found by adding 5 to 3/4 of the previous term. If the first term of the sequence is n and n ≠ 0, what is the ratio of the second term to the first term?
F. (n + 5)/4
G. (n + 20)/4
H. (3n + 5)/4n
J. (3n + 20)/4n
K. (5n + 3)/4n
You could try to solve this using variable expressions, but you’ll end up with a complex fraction (a fraction within a fraction). That doesn’t seem like fun! Instead, pick a number for n and try the problem that way. When you’re done, take a look at the solution below.
Let’s start by picking a first term, n. Since we know that we’ll have to take 3/4 of this number, it makes sense to choose a multiple of 4, so how about we try n = 8. The rest of our work would look something like this:
First term: 8
Second term: 5 + (3/4)(8) = 11
Ratio of second term to first term: 11/8
Now we’ll plug our value for n into each of the answer choices until one of them is equal to 11/8.
F. (8 + 5)/4 = 13/4
G. (8 + 20)/4 = 28/4 = 7
H. (3·8 + 5)/4·8 = 29/32
J. (3·8 + 20)/4·8 = 44/32 = 11/8 ← Eureka!
K. (5·8 + 3)/4·8 = 43/32
Choice J. works out to be 11/8 and that’s the correct answer!
Some Other Important Points:
- If you see variables in the answer choices, there’s a really good chance that problem will be a good one to try picking numbers.
- Percent increase/decrease problems also lend themselves well to picking numbers. You’ve seen them before. They start something like “The width of a rectangle is increased by 30 % …..” For these problems a good starting value is 100 since “percent” literally means “per hundred.”
- Avoid choosing 0 or 1 as these two numbers have special properties. For instance, multiplying a number by 1 and dividing that same number by 1 leaves you with the exact same value.
- Make sure you test all of the answer choices. It doesn’t happen often but there’s a chance that the numbers you choose will lead more than one of the answer choices to equal the value you’re looking for. When that happens, choose a new number to break the “tie.”
To try a few more of these problems, go here.
Questions about this strategy or anything else related to the ACT? Leave a comment or send me an email at firstname.lastname@example.org