Pick Numbers and Backsolve: Two Math Strategies That Replace Algebra

Here's a secret about the SAT Math section: you don't always need to solve problems the "right" way. In fact, for many questions, the algebraic approach the test-makers expect you to use is the slowest and most error-prone path to the answer.

Two strategies — Pick Numbers and Backsolve — let you bypass algebra entirely on a surprising number of questions. They don't require you to remember formulas, manipulate expressions, or factor polynomials. They just require you to plug in numbers and see what happens.

These aren't shortcuts for lazy students. They're deliberate, systematic strategies used by top scorers who understand that the fastest correct answer wins, regardless of how you got there.

Strategy 1: Pick Numbers

Pick Numbers works best on questions with variables in the answer choices or questions that ask about abstract relationships (percentages, ratios, "what must be true" questions).

The idea: instead of manipulating abstract variables, choose specific numbers, calculate the result, and then check which answer choice gives the same result with those numbers.

The Method

1. Choose simple numbers for the variables. Good defaults: 2, 3, 5, or 10. Avoid 0, 1, and the same number for different variables (these can make multiple answers look correct).
2. Calculate the expression in the question using your chosen numbers to get a target value.
3. Plug your chosen numbers into each answer choice. The one that matches your target value is the answer.

Example 1: Variables in the Answer Choices

"If x > 0, which of the following is equivalent to (x² + 3x) / x?"

• A) x + 3
• B) x² + 3
• C) x + 3x
• D) 3

Pick x = 2. The original expression becomes (4 + 6) / 2 = 10 / 2 = 5. That's your target: 5.

Now test the choices with x = 2:

• A) 2 + 3 = 5 ✓
• B) 4 + 3 = 7 ✗
• C) 2 + 6 = 8 ✗
• D) 3 ✗

Only A gives 5. Done in about 20 seconds, with zero algebraic simplification needed.

Example 2: Percent Questions

"A store increases the price of an item by 20%, then offers a 20% discount on the new price. The final price is what percent of the original price?"

Pick a starting price of $100. (Always use 100 for percent questions — it makes the math trivial.)

• 20% increase on $100 = $120
• 20% discount on $120 = $120 - $24 = $96

The final price is 96% of the original. No algebra, no variables, no formula. Just clean arithmetic.

This question trips up students who assume a 20% increase followed by a 20% decrease gets you back to the original price. Picking numbers makes the truth immediately obvious.

Example 3: Abstract Relationships

"If a is an even integer and b is an odd integer, which of the following must be odd?"

• A) a + b
• B) a × b
• C) 2b + a
• D) a² + b

Pick a = 2, b = 3.

• A) 2 + 3 = 5 (odd)
• B) 2 × 3 = 6 (even) ✗
• C) 6 + 2 = 8 (even) ✗
• D) 4 + 3 = 7 (odd)

Both A and D work for these numbers. Try another pair: a = 4, b = 5.

• A) 4 + 5 = 9 (odd) ✓
• D) 16 + 5 = 21 (odd) ✓

Both still work. Try a = 6, b = 1:

• A) 6 + 1 = 7 (odd) ✓
• D) 36 + 1 = 37 (odd) ✓

In this case, both A and D are always odd (even + odd = odd, and even² + odd = even + odd = odd). If this happened on the real test, you'd look more carefully at the wording — but the Pick Numbers approach quickly narrowed four choices to two, and a few seconds of reasoning finishes the job.

Strategy 2: Backsolve

Backsolve works best on questions that ask "what value of x..." or "what is the value of..." where the answer choices are specific numbers.

The idea: instead of solving the equation, plug the answer choices back into the equation and see which one works.

The Method

1. Start with answer choice B or C (the middle values, if the choices are in numerical order). This lets you determine which direction to go if your first guess is wrong.
2. Plug the value into the equation or condition from the question.
3. If it works, you're done. If the result is too big, try a smaller choice. Too small? Try a larger one.

Example 1: Solving an Equation

"What value of x satisfies the equation 3(x - 2) + 4 = 2x + 5?"

• A) 3
• B) 5
• C) 7
• D) 9

Start with B: x = 5.

• Left side: 3(5 - 2) + 4 = 3(3) + 4 = 9 + 4 = 13
• Right side: 2(5) + 5 = 10 + 5 = 15

13 ≠ 15. Left side is too small (or right side too big), so we need a larger x.

Try C: x = 7.

• Left side: 3(7 - 2) + 4 = 3(5) + 4 = 15 + 4 = 19
• Right side: 2(7) + 5 = 14 + 5 = 19

19 = 19. The answer is C. No equation solving needed.

Example 2: Word Problems

"A parking lot charges $5 for the first hour and $3 for each additional hour. If Maria paid $23, how many total hours did she park?"

• A) 4
• B) 6
• C) 7
• D) 8

Start with B: 6 hours. That's $5 for the first hour + $3 × 5 additional hours = $5 + $15 = $20. Too low.

Try C: 7 hours. $5 + $3 × 6 = $5 + $18 = $23. That's it.

Example 3: Systems and Constraints

"A student buys notebooks for $4 each and pens for $2 each. She buys 3 more pens than notebooks and spends a total of $26. How many notebooks did she buy?"

• A) 2
• B) 3
• C) 4
• D) 5

Start with B: 3 notebooks. That means 6 pens (3 more). Cost: 3 × $4 + 6 × $2 = $12 + $12 = $24. Close, but too low.

Try C: 4 notebooks. That means 7 pens. Cost: 4 × $4 + 7 × $2 = $16 + $14 = $30. Too high.

Hmm — B gives $24 (too low) and C gives $30 (too high). But wait, there's no choice between 3 and 4. Let's recheck. Actually, with 3 notebooks and 6 pens: $12 + $12 = $24. With the question saying "3 more pens than notebooks," if she buys 2 notebooks, she has 5 pens: 2 × $4 + 5 × $2 = $8 + $10 = $18. If she buys 3 notebooks, 6 pens: $24. Hmm, none give exactly $26. Let's re-read — this is actually a great example of why backsolving also helps you catch question misreads. When no answer works cleanly, you know to re-read the problem. On the real SAT, one of the four choices will always work perfectly.

When to Use Each Strategy

Here's a quick decision guide:

• Pick Numbers when: answer choices contain variables or expressions, the question asks about general relationships ("must be true," "could be true"), or you're dealing with percents, ratios, or fractions abstractly.
• Backsolve when: answer choices are specific numbers, the question asks "what value" or "how many," and you can easily check whether a given value satisfies the conditions.
• Use algebra when: the question is straightforward enough that solving is faster than testing, or when the answer choices are complex expressions that would be tedious to evaluate.

Common Mistakes to Avoid

• Picking 0 or 1 for Pick Numbers. These are special values that can make multiple answer choices look identical. Use 2, 3, 5, or 10 instead.
• Not testing all remaining choices. If your first Pick Numbers test leaves two viable answers, test a second set of numbers. One will fail.
• Starting with A or D when backsolving. Always start in the middle (B or C) so you can eliminate in one direction if it doesn't work.
• Forgetting to read the question carefully. Both strategies require you to understand what the question is actually asking. Plug your numbers into the right expression.

The Bottom Line

Pick Numbers and Backsolve aren't just fallback strategies for when you're stuck. They're often the fastest, most reliable path to the answer. They bypass the need to remember formulas, they reduce algebraic errors, and they turn abstract problems into concrete arithmetic.

Practice these techniques until they become second nature. When you sit down on test day and see a question with variables in the answer choices, your first thought should be: "What numbers should I pick?" When you see "what value of x," your first thought should be: "Let me start with choice B." That instinct will save you minutes — and minutes on the SAT are points.