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GMAT Quantitative Reasoning

GMAT algebra practice questions

GMAT Focus Quantitative Reasoning tests arithmetic and algebra, and its algebra questions reward speed with the standard moves: factorising, substituting and isolating. These ten original questions cover equations, absolute value, inequalities, functions and sequences. Try each before opening the solution.

Key ideas

  • Spot difference of squares and common factors before you expand anything.
  • An absolute value equation usually has two cases, so test both.
  • Multiplying or dividing an inequality by a negative number reverses the sign.
  • For function composition, work from the inside out.

Common traps

  • Forgetting the second case of an absolute value equation.
  • Not reversing an inequality sign after dividing by a negative number.
  • Cancelling terms from a sum as if they were factors.

Practice questions with solutions

Try each question first, then open the solution. Aim for about two minutes per question.

  1. If 3x + 2y = 16 and x - y = 2, what is the value of x + y?

    • A 8
    • B 6
    • C 4
    • D 12
    • E 10
    Show answer and solution

    Answer: B. 6

    From x - y = 2, x = y + 2. Then 3(y + 2) + 2y = 16, so 5y = 10, y = 2, x = 4. So x + y = 6.

  2. What is the sum of all values of x that satisfy |2x - 3| = 7?

    • A 3
    • B -3
    • C 10
    • D 5
    • E 7
    Show answer and solution

    Answer: A. 3

    2x - 3 = 7 gives x = 5. 2x - 3 = -7 gives x = -2. The sum is 3.

  3. How many integers x satisfy |x - 2| < 4?

    • A 6
    • B 7
    • C 4
    • D 9
    • E 8
    Show answer and solution

    Answer: B. 7

    |x - 2| < 4 means -2 < x < 6. The integers are -1, 0, 1, 2, 3, 4, 5, which is 7 integers.

  4. If x² - y² = 45 and x + y = 9, what is the value of x - y?

    • A 36
    • B 4
    • C 9
    • D 3
    • E 5
    Show answer and solution

    Answer: E. 5

    x² - y² = (x + y)(x - y). So 45 = 9(x - y), and x - y = 5.

  5. The roots of x² - 7x + 10 = 0 are a and b. What is the value of a² + b²?

    • A 21
    • B 49
    • C 39
    • D 29
    • E 25
    Show answer and solution

    Answer: D. 29

    The roots are 2 and 5, since (x - 2)(x - 5) = 0. So a² + b² = 4 + 25 = 29. You can also use (a + b)² - 2ab = 49 - 20 = 29.

  6. If f(x) = 2x + 3 and g(x) = x², what is the value of g(f(1))?

    • A 25
    • B 36
    • C 10
    • D 16
    • E 17
    Show answer and solution

    Answer: A. 25

    f(1) = 5. Then g(5) = 25. Note the order: g(f(1)) applies f first.

  7. If 2^x = 32 and 3^y = 81, what is the value of xy?

    • A 20
    • B 12
    • C 9
    • D 15
    • E 24
    Show answer and solution

    Answer: A. 20

    2^5 = 32, so x = 5. 3^4 = 81, so y = 4. Then xy = 20.

  8. If (x + 1)/(x - 2) = 3, what is x?

    • A 5/2
    • B -3/2
    • C 4
    • D 7/2
    • E 3
    Show answer and solution

    Answer: D. 7/2

    Multiply both sides by (x - 2): x + 1 = 3x - 6, so 7 = 2x and x = 7/2.

  9. What is the largest integer x such that 5 - 2x > 11?

    • A -3
    • B 3
    • C -2
    • D -6
    • E -4
    Show answer and solution

    Answer: E. -4

    5 - 2x > 11 gives -2x > 6. Dividing by -2 reverses the sign: x < -3. The largest integer below -3 is -4.

  10. The first term of an arithmetic sequence is 3 and the common difference is 4. What is the sum of the first 20 terms?

    • A 880
    • B 820
    • C 840
    • D 780
    • E 800
    Show answer and solution

    Answer: B. 820

    The 20th term is 3 + 19 x 4 = 79. The sum is 20 x (3 + 79) / 2 = 10 x 82 = 820.

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