One extremely effective strategy in tackling tough questions is making up numbers and testing each answer choice.

To see how this works, here are three examples:

A) 20%

B) 50%

C) 65%

D) 75%

Percent questions are often perfect candidates for this strategy. In these cases, the number \(100\) is usually a great number to work with.

So let's pretend the initial price of the dress is $100.

After a 30% markup, the new price is $130.

Now we mark it down by half (50%): $65.

The final price is $65, which is 65% of the original price. The answer is **C**.

Not complicated at all, right? Let's use this technique to solve a much harder problem.

2. If \(n\) is a positive integer and \(3^n + 3^{n+1} = m\), what is \(3^{n+2}\) in terms of \(m\)?

A) \(\dfrac{m-1}{2}\)

B) \(\dfrac{9m}{4}\)

C) \(3m\)D) \(3m + 1\)

Even if you have no idea what's going on here, you can still answer the question by making up numbers.

Let's pretend that \(n = 1\), a nice simple number. Then,

\[m = 3^1 + 3^2 = 12\]Also,

\[3^{n+2} = 3^3 = 27\]Now all we have to do is plug our value of \(m = 12\) into each of the answer choices to see which one gives \(27\).

The only answer choice that gives us \(27\) is **B**.

When using this strategy, you should ALWAYS review every answer choice, even when you think you've found the answer. The reason is that another answer choice may have also given us \(27\). When two answer choices give the desired result, make up another number (e.g. \(n=2\)) and repeat the process with just those two answer choices. This will filter out the answer choice that just happens to work in one case but not the other. You want the answer that works in all cases.

It may sound like a long process, but running through the answer choices typically doesn't take long at all, especially with a calculator. And most of the time, you won't run into a situation where you have to make up more than one number to test the answers with.

Ah, another perfect question for making up numbers. Solving this question "the standard way" would be a huge pain. Let's guess a bit. Let's try \(b = 2\).

\(f(2) = \left|4(2) - 25\right| = \left|-17\right| = 17\), which is not less than \(b\) itself. So \(b\) cannot be 2.

Let's try another value. Maybe \(b = 5\).

\(f(5) = \left|4(5) - 25\right| = \left|-5\right| = 5\), which is not less than \(b\) itself. So \(b\) cannot be 5.

But we're a lot closer!! When \(b = 5\), \(f(b)\) is also equal to \(5\).

So let's try something larger, \(b = 6\).

\(f(6) = \left|4(6) - 25\right| = \left|-1\right| = 1\), and that works!! \(f(6) \lt 6\).

So one possible answer to this question is **6**.

Hopefully, you have an idea of how this works. Whenever you get stuck, it's good to have this strategy in mind. It's likely to be useful in questions dealing with:

- Percent
- Ratios or Proportions
- Questions with variables (like in #2 and #3 above)

The more you practice, the better you'll get at recognizing when this strategy is the way to go. So with that, I leave you with a bunch of practice questions suited to this technique. Try out this strategy even if you know how to solve the question mathematically!

#### Practice Questions

- If \(\left|x\right| = \left|y\right|\) and \(x \neq y\), which of the following must be true?

- \(x - y = 0\)
- \(xy > 0\)
- \(x^2 = y^2\)

B) III only

C) I and II only

D) I, II, and III - If \(k > 1\), what is the slope of the line in the \(xy\)-plane that passes through the points \((k,k^2)\) and \((k^2,k^3)\)?

A) \(k\)

B) \(k^2\)

C) \(k^2 + k\)

D) \(k^2 - k\) - When the positive integer \(x\) is divided by 10, the remainder is 4. When the positive integer \(y\) is divided by 10, the remainder is 7. What is the remainder when \(x+y\) is divided by 10?

A) 1

B) 2

C) 3

D) 4 - For nonzero numbers, \(a\), \(b\), \(c\), if \(a\) is twice \(b\) and \(b\) is \(\dfrac{1}{4}\) of \(c\), what is the ratio of \(a^2\) to \(c^2\)?

A) 1 to 2

B) 1 to 4

C) 1 to 8

D) 1 to 16 - If \(k\) is a two-digit number whose units digit is the same as its tens digit, which of the following statements must be true?

A) \(k\) is a multiple of 2

B) \(k\) is a multiple of 5

C) \(k\) is a multiple of 11

D) \(k\) is greater than 30 - If \(a\) is an even integer and \(b\) is an odd integer, which of the following is an even integer?

A) \(a^2 + b^2\)

B) \(a^2 + b^3\)

C) \(a^3 + b^2\)

D) \(a^2 + ab\) - In a book collection, 50% of the books have 100 pages, 40% have 150 pages, and 10% have 200 pages. What is the average (arithmetic mean) number of pages per book?

A) 120

B) 125

C) 130

D) 135

#### Answers

- B
- A
- A
- B
- C
- D
- C