Log inGet started
DrillsLessonsSectionsPractice testsPricingTry 5 questions
← All lessonsMATH · TECHNIQUE

Picking numbers

Replace variables or unknown amounts with easy numbers and compare the choices, instead of proving something in general.

When to use it

  • The choices contain variables and the question asks which expression is equal, must be true, or gives the result.
  • A percent or fraction problem never gives the starting amount.
  • The question says “must be even/odd” or “always true”.

How it works

  1. Choose small, simple numbers that respect any conditions (odd, positive, x ≠ 2). For percents use 100. Avoid 0, 1 and numbers that make two choices look alike.
  2. Compute the target with your numbers.
  3. Compute every choice with the same numbers and keep only the ones that match.
  4. If two choices still match, pick a second set of numbers. Different numbers separate impostors.
Watch out: Stopping after one number. “Must be true” needs a choice that survives every legal number you try, and one test can leave two choices standing.

Time: Two quick numbers beat five lines of algebra if the algebra is long. If a choice survives both numbers, it is almost certainly the answer.

Worked examples

Example 1

No starting price is given, so pick one.

A price is increased by 20% and then decreased by 20%. What is the overall change from the original price?

  1. ANo change
  2. BA 4% increase
  3. CA 20% decrease
  4. DA 4% decreaseCorrect

Using the technique

  1. Choose a price of $100.

  2. Up 20%: 100 × 1.2 = $120. Down 20% of the new price: 120 × 0.8 = $96.

  3. $96 is $4 less than $100, a 4% decrease. Percent changes do not cancel because the second one is taken on a different base.

Practice this set

Example 2

“Must be even” for any odd n: test odd numbers.

If n is an odd integer, which of the following must be even?

  1. An² + 2
  2. Bn + 2
  3. Cn² + nCorrect
  4. D3n

Using the technique

  1. Let n = 3. The choices become n² + 2 = 11, n + 2 = 5, n² + n = 12 and 3n = 9.

  2. Only n² + n is even. Confirm with a second odd number, n = 1: the choices give 3, 3, 2 and 3, and again only n² + n is even.

  3. It survives both tests, so it is the answer. (Algebra agrees: n² + n = n(n + 1), a product of consecutive integers.)

Practice this set

Example 3

One number may not separate all the choices. Show what to do when it does not.

Which expression is equivalent to (x² − 5x + 6)/(x² − 4), for x ≠ ±2?

  1. A(x + 3)/(x − 2)
  2. B(x − 3)/(x − 2)
  3. C(x − 3)/(x + 2)Correct
  4. D−5x/(−4)

Using the technique

  1. Try x = 0. The original is 6/(−4) = −1.5. Choice A gives 3/(−2) = −1.5 and choice C gives (−3)/2 = −1.5. Both match, B gives 1.5 and D gives 0, so B and D are out.

  2. Two choices are left, so use a second number. Try x = 1 (allowed, since x ≠ ±2). The original is (1 − 5 + 6)/(1 − 4) = 2/(−3).

  3. Choice A gives 4/(−1) = −4, which fails. Choice C gives (−2)/3 = −2/3, which matches. The answer is C.

Practice this set

Practice with a custom Math set

Related lessons: Percents · Rational expressions · Number properties