Topic 2

Picking Numbers for Variable Expressions

Picking numbers is a strategy for problems that ask about general relationships between variables rather than specific numeric answers. When a problem says ``for all values of

x

'' or asks which expression is equivalent to another, you can choose a convenient value for the variable, evaluate both sides, and see which option matches.

This strategy is especially useful for:

Equivalent expressions: ``Which of the following is equivalent to $\dfrac{x^2 - 4}{x + 2}$ ?'' Pick $x = 3$ : original $= \dfrac{9 - 4}{5} = 1$ . Check whether each candidate also gives 1 when $x = 3$ .
Percent and ratio problems with variables: ``If the price of an item is increased by $p\%$ , then decreased by $p\%$ , the final price is what fraction of the original?'' Pick $p = 20$ (or any convenient value) and compute.
``Must be true'' questions: ``If $n$ is a positive even integer, which of the following must be odd?'' Test with $n = 2$ and $n = 4$ to rule out options.

Rules for picking numbers well:

Avoid 0 and 1. These are special cases that can make different expressions look equal. For instance, $x$ and $x^2$ are both 1 when $x = 1$ .
Pick numbers that make the arithmetic easy. For percent problems, 100 is a great starting value. For fractions, pick a common denominator.
Respect constraints. If the problem says $x > 0$ , do not pick $x = -1$ . If it says $x$ is odd, pick an odd number.
If two candidates match, try a second number. Picking one number might not distinguish between two expressions. A second test usually resolves it.

On the Digital SAT's free-response format, picking numbers is most useful as a verification tool: solve the problem algebraically, then check your answer by substituting a chosen value into both the original expression and your answer.

Core Skills

Choose strategic test values that make arithmetic simple while avoiding special cases.
Evaluate algebraic expressions with substituted values to verify equivalence.
Use picked numbers to test general percent and ratio relationships.
Distinguish between ``must be true,'' ``could be true,'' and ``must be false'' by testing multiple values.

Example 1: Simplifying an Expression

Simplify

\dfrac{x^2 - 9}{x - 3}

for

x \neq 3

. Algebraic approach: Factor the numerator:

\dfrac{(x-3)(x+3)}{x-3} = x + 3

. Verification by picking numbers: Let

x = 5

. Original:

\dfrac{25 - 9}{2} = 8

. Simplified:

5 + 3 = 8

. Matches.

\boxed{x + 3}

Example 2: Percent Problem with Variables

A shirt's price is

d

dollars. It is marked up by

20\%

and then discounted by

10\%

. What is the final price in terms of

d

? Picking numbers: Let

d = 100

. After

20\%

markup:

120

. After

10\%

discount:

120 \times 0.90 = 108

. So the final price is

1.08d

. Algebraic confirmation:

d \times 1.20 \times 0.90 = 1.08d

\boxed{1.08d}

Example 3: ``Must Be True'' Problem

n

is a positive integer, which expression must be even:

2n + 1

n^2 + n

, or

n + 3

? Test $n = 2$:

2(2) + 1 = 5

(odd),

4 + 2 = 6

(even),

2 + 3 = 5

(odd). Test $n = 3$:

7

(odd),

9 + 3 = 12

(even),

6

(even). Only

n^2 + n = n(n+1)

is even for both. This makes sense:

n(n+1)

is the product of consecutive integers, so one factor is always even.

\boxed{n^2 + n}

Example 4: Picking Numbers to Verify an Algebraic Result

Simplify

\dfrac{2(x + 3) - 4}{2}

. Algebraic:

\dfrac{2x + 6 - 4}{2} = \dfrac{2x + 2}{2} = x + 1

. Check with $x = 4$: Original:

\dfrac{2(7) - 4}{2} = \dfrac{10}{2} = 5

. Simplified:

4 + 1 = 5

. Confirmed.

\boxed{x + 1}

Key Takeaways

Picking numbers turns abstract algebra problems into concrete arithmetic. It is the fastest way to check whether an algebraic simplification is correct.
Always avoid 0 and 1 unless the problem forces them. These values cause too many expressions to coincide.
For percent problems with no specific starting value, start with 100. It turns every percent directly into a dollar amount.
Picking numbers is a verification strategy on free-response questions. You still need algebra to produce a candidate answer, but picking numbers confirms it in seconds.