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Grade 9 math review pack

30 mixed questions from every topic in Grade 9, with an answer key and explanations. Great for the start of the year, before a test, or over the holidays.

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Grade 9 Math Review Pack

Name: ____________________Date: ____________Score: ____ / 30
  1. Integers
    Work out $|-1| - |-3| \times 1$.
    Answer: ______________________
  2. Algebra
    Solve for x: $3(x + 5) = 2x + 8$
    Answer: ______________________
  3. How many integers x satisfy $-2 \le 4x + 5 < 20$?
    Answer: ______________________
  4. Expand and simplify $2x(4x + 3) - 3(2x^{2} - 3x - 1)$.
    A. $4x^{2} + 15x + 3$B. $2x^{2} + 15x + 1$C. $2x^{2} + 15x + 3$D. $2x^{2} + 9x + 3$
  5. Solve for x: $\dfrac{x}{5} - 8 = -12$
    Answer: ______________________
  6. Exponents
    Solve for x: $16^{x + 3} \times 8 = 8^{x + 4}$
    Answer: ______________________
  7. Write $x^{\dfrac{1}{3}} \times x^{\dfrac{1}{5}}$ in simplest form.
    A. $x^{\dfrac{2}{8}}$B. $x^{\dfrac{1}{8}}$C. $x^{\dfrac{1}{15}}$D. $x^{\dfrac{8}{15}}$
  8. Write $\dfrac{2x^{-2}y}{2x^{4}y^{-2}}$ in simplest form. Use positive exponents only.
    A. $\dfrac{y^{3}}{x^{6}}$B. $\dfrac{y^{5}}{x^{6}}$C. $\dfrac{y}{x^{6}}$D. $\dfrac{y^{3}}{x^{8}}$
  9. Radicals
    Simplify $\dfrac{3}{1 + \sqrt{7}} + \dfrac{2}{\sqrt{7} - 4}$.
    A. $\dfrac{-25 + 5\sqrt{7}}{18}$B. $\dfrac{25 + 5\sqrt{7}}{18}$C. $\dfrac{15 + 10\sqrt{7}}{19}$D. $\dfrac{7 + 13\sqrt{7}}{18}$
  10. Rationalize the denominator: $\dfrac{5 + \sqrt{7}}{4 + 3\sqrt{7}}$
    A. $-\dfrac{41 + 19\sqrt{7}}{47}$B. $\dfrac{1 + 11\sqrt{7}}{47}$C. $-\dfrac{1 + 11\sqrt{7}}{47}$D. $\dfrac{-1 + 11\sqrt{7}}{47}$
  11. Simplify: $5\sqrt{405}$.
    A. $45\sqrt{5}$B. $45\sqrt{10}$C. $46\sqrt{5}$D. $-45\sqrt{5}$
  12. Rates, Percent and Finance
    A recipe uses 6 cups of flour for 4 muffins. How many muffins can be made with 48 cups?
    Answer: ______________________
  13. A watch was sold at a 5% loss. If it had been sold for $\$118$ more, there would have been a 5% profit. What was the cost price? (dollars)
    Answer: ______________________
  14. After a 60% increase, a price is $\$160$. What was the original price? (dollars)
    Answer: ______________________
  15. Sequences
    Which term of $6,\; 3,\; 0,\; \dots$ is $-153$? (Give n.)
    Answer: ______________________
  16. Find term 6 of $-5,\; 15,\; -45,\; \dots$
    Answer: ______________________
  17. Linear Relations
    Which equation matches the table?
    A. $y = 3x + 20$B. $y = -3x + 17$C. $y = 20x - 3$D. $y = -3x + 20$
  18. Which equation matches the line on the graph?
    A. $y = 4x - 4$B. $y = -\dfrac{1}{4}x - 4$C. $y = -4x - 4$D. $y = -4x + 4$
  19. What is the slope of a line perpendicular to $y = -\dfrac{1}{3}x - 5$?
    Answer: ______________________
  20. Quadratic Relations
    Find the equation of the axis of symmetry of $y = -2(x + 1)(x - 1)$. Give the value of h in x = h.
    Answer: ______________________
  21. A rectangle's length is 1 m more than its width. Its area is 56 m². What is the width? (m)
    Answer: ______________________
  22. Write $y = 2(x + 6)^2 + 0$ in standard form.
    A. $y = 2x^2 + 72$B. $y = 2x^2 + 24x + 72$C. $y = 2x^2 - 24x + 72$
  23. Trigonometry
    In a right triangle, $\sin A = \dfrac{24}{25}$. What is $\tan A$? (Give a fraction.)
    Answer: ______________________
  24. What is the exact value of $\tan 45^\circ$?
    A. $\dfrac{1}{2}$B. 1C. $\dfrac{\sqrt{3}}{2}$D. $\sqrt{3}$
  25. A 5 m ladder makes an angle of 66° with the ground. How high up the wall does it reach, to the nearest tenth? (m)
    Answer: ______________________
  26. Measurement and Geometry
    How many yards are in 7 miles?
    Answer: ______________________
  27. Work out the volume of a cone with radius 9 cm and height 14 cm, to the nearest tenth. (cm³)
    Answer: ______________________
  28. Find the area of a 240° sector of a circle with radius 10 cm, to the nearest tenth.
    Answer: ______________________
  29. Data
    Work out the first quartile (Q1) of: 18, 18, 20, 21, 24, 27, 37, 42, 53, 53, 53.
    Answer: ______________________
  30. A line of best fit for hours studied (x) and test score (y) is $y = 6x + 37$. Predict the score for 3 hours.
    Answer: ______________________

Answer key

  1. -2 — |−1| = 1 and |−3| = 3. Multiply first: 3 × 1 = 3. Then 1 − (3) = −2.
  2. -7 — Expand: 3x + 15 = 2x + 8. Collect: 1x = −7, so x = −7.
  3. 5 — Subtract 5 from all three parts, then divide by 4: -1.75 ≤ x < 3.75. The integers in this range are −1, 0, 1, 2, 3: that's 5. (If x could be any real number, there would be infinitely many solutions.)
  4. C ($2x^{2} + 15x + 3$) — First: $2x(4x + 3) = 8x^{2} + 6x$. Second: $-3(2x^{2} - 3x - 1) = -6x^{2} + 9x + 3$ (the minus multiplies every term!). Collect like terms: $2x^{2} + 15x + 3$.
  5. -20 — Subtract −8 on both sides: x/5 = −4. Multiply by 5: x = −20.
  6. -3 — Powers of 2: $2^{4(x + 3)} \times 2^{3} = 2^{3(x + 4)}$. Add exponents on the left: 4(x + 3) + 3 = 3(x + 4). Solve: x = −3.
  7. D ($x^{\dfrac{8}{15}}$) — Same base, so add the exponents: $\dfrac{1}{3} + \dfrac{1}{5} = \dfrac{8}{15}$.
  8. A ($\dfrac{y^{3}}{x^{6}}$) — Subtract exponents: x: −2 − 4 = −6; y: 1 − (−2) = 3. Move negative exponents to the denominator: $\dfrac{y^{3}}{x^{6}}$.
  9. A ($\dfrac{-25 + 5\sqrt{7}}{18}$) — Rationalize each fraction on its own. $\dfrac{3}{1 + \sqrt{7}} = \dfrac{-1 + \sqrt{7}}{2}$ (conjugate $1 - \sqrt{7}$, bottom -6). $\dfrac{2}{\sqrt{7} - 4} = -\dfrac{8 + 2\sqrt{7}}{9}$ (conjugate $\sqrt{7} + 4$, bottom -9). Add: $\dfrac{-25 + 5\sqrt{7}}{18}$.
  10. B ($\dfrac{1 + 11\sqrt{7}}{47}$) — Multiply top and bottom by the conjugate $4 - 3\sqrt{7}$. Bottom: $4^2 - (3\sqrt{7})^2 = 16 - 63 = -47$. Top: $\left(5 + \sqrt{7}\right)\left(4 - 3\sqrt{7}\right)$ = $-1 - 11\sqrt{7}$. Answer: $\dfrac{1 + 11\sqrt{7}}{47}$.
  11. A ($45\sqrt{5}$) — Find the biggest perfect square factor: 405 = 81 × 5. $\sqrt{81} = 9$, so $5\sqrt{405} = 5 \times 9\sqrt{5} = 45\sqrt{5}$.
  12. 32 — Set up 6/4 = 48/m → m = 4 × 48 ÷ 6 = 32.
  13. 1180 — The difference between a 5% loss and a 5% profit is 10% of the cost. 10% of cost = 118, so cost = 118 ÷ 0.1 = 1180.
  14. 100 — The new price is 160% of the original: 160 ÷ 1.6 = 100.
  15. 54 — −153 = 6 + (n − 1)(−3) → n − 1 = 53 → n = 54.
  16. 1215 — tₙ = a × r^(n − 1) = −5 × (−3)^5 = 1215.
  17. D ($y = -3x + 20$) — Rate of change −3 (the slope), initial value 20 (y when x = 0): y = −3x + 20.
  18. C ($y = -4x - 4$) — Read the y-intercept: −4. From (0, −4), go 1 right and 4 down to the next point, so the slope is −4.
  19. 3 — Perpendicular slopes are negative reciprocals: flip −1/3 and change the sign → 3. (Their product is −1.)
  20. 0 — The axis of symmetry is halfway between the zeros −1 and 1: x = (−1 + 1) ÷ 2 = 0.
  21. 7 — Let the width be w: w(w + 1) = 56 → w² + 1w − 56 = 0 → (w − 7)(w + 8) = 0. A width can't be negative, so w = 7 m.
  22. B ($y = 2x^2 + 24x + 72$) — (x + 6)² = x² + 12x + 36. Multiply by 2 and add 0: $y = 2x^2 + 24x + 72$.
  23. 24/7 — sin A = opposite/hypotenuse, so opposite = 24 and hypotenuse = 25. By Pythagoras the adjacent side is √(25² − 24²) = 7. tan A = 24/7.
  24. B (1) — Use the special triangles: in a 30-60-90 triangle the sides are 1, √3, 2; in a 45-45-90 triangle they are 1, 1, √2. tan 45° = 1.
  25. 4.6 — sin 66° = height ÷ 5, so height = 5 × sin 66° ≈ 4.6 m.
  26. 12320 — 1 mile = 1760 yards: 7 × 1760 = 12320.
  27. 1187.5 — V = ⅓πr²h = ⅓ × π × 81 × 14 ≈ 1187.5 cm³. (A cone holds exactly one third of the cylinder with the same base and height.)
  28. 209.4 — Sector = (240/360) × π × 10² ≈ 209.4 cm².
  29. 20 — The median is 27. Q1 is the median of the lower half (18, 18, 20, 21, 24): 20.
  30. 55 — y = 6 × 3 + 37 = 55.

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