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Grade 7 math review pack

30 mixed questions from every topic in Grade 7, 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 7 Math Review Pack

Name: ____________________Date: ____________Score: ____ / 30
  1. Rational Numbers
    $-\dfrac{1}{5} - \left(\dfrac{1}{3}\right)$ = ? Simplify.
    Answer: ______________________
  2. Which is the least?
    A. $\dfrac{3}{7}$B. 8C. $\dfrac{4}{7}$D. $-\dfrac{1}{2}$
  3. Exponents and Scientific Notation
    $(1.8 \times 10^{8}) \times (4.5 \times 10^{5})$ = ?
    A. $81 \times 10^{13}$B. $8.1 \times 10^{13}$C. $8.1 \times 10^{40}$D. $8.1 \times 10^{14}$
  4. A square garden bed has an area of 10.89 m². How long is each side? (m)
    Answer: ______________________
  5. Percent and Finance
    A city's population grew from 2,100 to 4,200. What was the percent increase?
    A. 10%B. 20%C. 100%D. 200%
  6. A restaurant meal costs $\$93.58$ before tax. 13% sales tax is added, then a 15% tip on the total with tax. What is the final amount?
    A. $\$120.15$B. $\$107.62$C. $\$121.61$D. $\$105.75$
  7. Ratios and Proportional Reasoning
    On a map with scale 1 : 100,000, how many cm apart are two towns that are really 3 km apart?
    Answer: ______________________
  8. Concrete is mixed as cement : sand : gravel = 4 : 6 : 3. How many kg of sand go with 52 kg of cement?
    Answer: ______________________
  9. Introduction to Algebra
    Tickets cost 5 dollars each plus a 12 dollar booking fee. Which expression gives the total cost, in dollars, for t tickets? What is it when t = 8?
    A. $5(t + 12)$; 100 dollarsB. $12t + 5$; 101 dollarsC. $5t + 12$; 52 dollarsD. $17t$; 136 dollars
  10. Algebra
    The volume of a box is $V = lwh$. A box has volume 320 cm³, length 4 cm and width 8 cm. Rearrange to find h. (cm)
    Answer: ______________________
  11. Solve: $\dfrac{x}{2} + \dfrac{x}{4} = 6$
    Answer: ______________________
  12. Linear Relations
    Gym A charges 35 dollars to join plus 20 dollars per visit. Gym B charges 77 dollars to join plus 18 dollars per visit. After how many visits do they cost the same?
    Answer: ______________________
  13. Is the point (4, −3) on the line $y = x - 5$?
    A. YesB. No
  14. Sequences and Series
    Which formula gives the nth term of −7, −15, −23, −31, …?
    A. $t_n = -8n + 1$B. $t_n = -7n - 8$C. $t_n = n - 8$D. $t_n = -8n - 7$
  15. The sequence 2, 4, 8, 16, … multiplies by the same number each time. What is term 7?
    Answer: ______________________
  16. Counting and Combinatorics
    How many arrangements of the letters in TOMATO end with the letter O?
    Answer: ______________________
  17. In how many ways can a gold, a silver and a bronze place be awarded among 18 runners?
    Answer: ______________________
  18. Factorials and Number Theory
    How many factors of 2 are in 54!? (That is, the exponent of 2 in the prime factorization of 54!.)
    Answer: ______________________
  19. What is the ones digit of $1! + 2! + 3! + \dots + 5!$?
    Answer: ______________________
  20. Geometry Theorems
    Which theorem says that vertically opposite angles are equal?
    A. OATB. TPT-ZC. ITTD. SAT
  21. Line segments AB and CD cross at E. ∠AEC = 112°. Find ∠AED. Which theorem gives the answer?
    A. SATTB. EATC. SATD. OAT
  22. Similar and Congruent Triangles
    Triangles ABC and DEF are congruent (A ↔ D, B ↔ E, C ↔ F). AB = 21 cm, DE = (2x + 11) cm, BC = 20 cm and EF = (3y + 11) cm. Find x + y.
    Answer: ______________________
  23. In triangle ABC, DE is parallel to BC, with D on AB and E on AC. AD = 8 cm, DB = 5 cm and DE = 6.5 cm. Find BC. (cm)
    Answer: ______________________
  24. Circles and Cylinders
    A running track has two straight sides 65 m long and two semicircular ends with diameter 57 m. What is the distance once around? (π ≈ 3.14)
    A. 244 mB. 487.96 mC. 243.98 mD. 308.98 m
  25. What is the total surface area of a closed cylinder with radius 14 cm and height 7 cm? (π ≈ 3.14)
    A. 4308.08 cm²B. 615.44 cm²C. 1846.32 cm²D. 1230.88 cm²
  26. Geometry
    An isosceles triangle has a top (apex) angle of 86°. What is each base angle? (degrees)
    Answer: ______________________
  27. The two horizontal lines are parallel. Find the angle marked x. (degrees)
    Answer: ______________________
  28. Contest Problems
    A circle of radius 2 cm fits exactly inside a square. What fraction of the square's area does the circle cover?
    A. $\dfrac{\pi}{2}$B. $\dfrac{1}{\pi}$C. $\dfrac{\pi}{4}$D. $\dfrac{3}{4}$
  29. What is the sum of the arithmetic sequence 6, 9, 12, …, 72?
    Answer: ______________________
  30. Data and Probability
    160 students were surveyed. In the circle graph the Art sector has an angle of 90°. How many students chose Art?
    Answer: ______________________

Answer key

  1. -8/15 — Common denominator 15: −3/15 − (5/15) = −8/15.
  2. D ($-\dfrac{1}{2}$) — As decimals: 8 ≈ 8, −1/2 ≈ −0.5, 3/7 ≈ 0.43, 4/7 ≈ 0.57. The most negative is −1/2.
  3. B ($8.1 \times 10^{13}$) — Multiply the numbers: 1.8 × 4.5 = 8.1. Add the exponents: 8 + 5 = 13.
  4. 3.3 — Side = √10.89. 3.3 × 3.3 = 10.89, so each side is 3.3 m.
  5. C (100%) — Increase: 4,200 − 2,100 = 2,100. ÷ original 2,100 = 1 = 100%. (The new population is 200% of the old, but the increase is 100%.)
  6. C ($\$121.61$) — With tax: 93.58 × 1.13 = 105.75. With tip: 105.75 × 1.15 = 121.61 dollars.
  7. 3 — 3 km = 300,000 cm. ÷ 100,000 = 3 cm on the map.
  8. 78 — 52 ÷ 4 = 13 per part. Sand: 6 × 13 = 78 kg.
  9. C ($5t + 12$; 52 dollars) — Each ticket adds 5, so 5t, plus the fee once: 5t + 12. When t = 8: 5 × 8 + 12 = 52.
  10. 10 — h = V ÷ (lw) = 320 ÷ (4 × 8) = 320 ÷ 32 = 10 cm.
  11. 8 — Multiply every term by the LCM 4: 2x + 1x = 24. So 3x = 24, x = 8.
  12. 21 — 35 + 20v = 77 + 18v. 2v = 42. v = 21 visits. (After that, Gym B is cheaper.)
  13. B (No) — Substitute x = 4: y = 1 × 4 − 5 = −1. That is not −3, so no.
  14. A ($t_n = -8n + 1$) — t_n = a + (n − 1)d = −7 + (n − 1)(−8) = −8n + 1. Check n = 1: −7 ✓.
  15. 128 — Common ratio r = 4 ÷ 2 = 2. Term n = a × r^(n − 1) = 2 × 2^6 = 128.
  16. 60 — Put one O at the end. Arrange the remaining 5 letters, dividing by the factorials of any repeats: 60.
  17. 4896 — Order matters: 18 × 17 × 16 = 4,896 ways.
  18. 50 — Count multiples of 2, then extra ones from 2², 2³, …: 54 ÷ 2 → 27, 54 ÷ 4 → 13, 54 ÷ 8 → 6, 54 ÷ 16 → 3, 54 ÷ 32 → 1. Total: 27 + 13 + 6 + 3 + 1 = 50. (Each multiple of 2 gives one 2; multiples of 2² give an extra one, and so on.)
  19. 3 — 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. From 5! on, every factorial ends in 0 (it contains 2 × 5). So the ones digit is 3.
  20. A (OAT) — OAT: vertically opposite angles are equal.
  21. C (SAT) — SAT: ∠AEC and ∠AED are on a straight line, so ∠AED = 180 − 112 = 68°.
  22. 8 — Corresponding sides are equal: 2x + 11 = 21 gives x = 5; 3y + 11 = 20 gives y = 3. x + y = 8.
  23. 10.56 — DE ∥ BC makes triangle ADE similar to triangle ABC. AB = 8 + 5 = 13. Scale factor = AB ÷ AD = 13 ÷ 8. BC = 6.5 × 13 ÷ 8 = 10.56 cm.
  24. D (308.98 m) — Two semicircles make one full circle: πd = 3.14 × 57 = 178.98. Plus the straights: 2 × 65 = 130. Total 308.98 m.
  25. C (1846.32 cm²) — Two circles: 2πr² = 1230.88. Curved side (a rectangle when unrolled): 2πr × h = 615.44. Total 1846.32 cm².
  26. 47 — The base angles are equal: (180 − 86) ÷ 2 = 47°.
  27. 76 — Alternate exterior angles are equal. x = 76°.
  28. C ($\dfrac{\pi}{4}$) — Square side = 4, area 16. Circle area = π × 4. Fraction = 4π ÷ 16 = π/4 (about 78.5%), for any size.
  29. 897 — There are (72 − 6) ÷ 3 + 1 = 23 terms. Sum = number of terms × (first + last) ÷ 2 = 23 × 78 ÷ 2 = 897.
  30. 40 — 90° out of 360° is 25%. 25% of 160 = 40 students.

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