Grade 6 Expressions & Equations Practice — Hard

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Question 1 of 23: Evaluate (a + b)(a − b) when a = 5 and b = 3.

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Answer key for parents & teachers (23 questions)
  1. Evaluate (a + b)(a − b) when a = 5 and b = 3. — 16. (5 + 3)(5 − 3) = (8)(2) = 16. (This is also a² − b² = 25 − 9 = 16.)
  2. Evaluate n² + 2n − 8 when n = −3. — −5. (−3)² + 2(−3) − 8 = 9 − 6 − 8 = −5.
  3. Which expression gives the same value as 3x² when x = 2? — 12. 3(2)² = 3(4) = 12. So the expression equals 12 when x = 2.
  4. Evaluate (2x)² − x² when x = 3. — 27. (2×3)² − 3² = (6)² − 9 = 36 − 9 = 27.
  5. Evaluate 10 − 3x² when x = −2. — −2. 10 − 3(−2)² = 10 − 3(4) = 10 − 12 = −2.
  6. Evaluate xy + x²y when x = 2 and y = 3. — 18. (2)(3) + (2)²(3) = 6 + 4(3) = 6 + 12 = 18.
  7. A rectangle has length (2x + 1) and width x. Write the expression for its perimeter. — 6x + 2. Perimeter = 2(length + width) = 2(2x + 1 + x) = 2(3x + 1) = 6x + 2.
  8. Solve: 10 − 3x = −2 — x = 4. 10 − 3x = −2 → −3x = −12 → x = 4.
  9. Solve for n: n/6 − 3 = 4 — n = 42. n/6 − 3 = 4 → n/6 = 7 → n = 42.
  10. Write an equation: "Three times a number plus 8 equals 29." Then solve. — 3n + 8 = 29; n = 7. "Three times a number" = 3n. "Plus 8 equals 29" → 3n + 8 = 29 → 3n = 21 → n = 7.
  11. When multiplying or dividing an inequality by a negative number, what must you do? — Flip the inequality sign. Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.
  12. Solve: −2x > 10 — x < −5. Divide both sides by −2 and FLIP the sign: x < 10/(−2) = −5. So x < −5.
  13. A customer pays $35 for a service plus $12 per hour. They paid $83 total. How many hours? — 4 hours. 35 + 12h = 83 → 12h = 48 → h = 4 hours.
  14. A car rental costs $25 per day plus a $40 fee. If the total cost is at most $215, how many days at most can you rent? — 7 days. 25d + 40 ≤ 215 → 25d ≤ 175 → d ≤ 7. At most 7 days.
  15. If the rate (constant of proportionality k) is 0.5, the equation y = kx gives: — y = 0.5x (y grows slower than x). k = 0.5 means y = 0.5x. Since k < 1, y grows more slowly than x. For every 2 units of x, y increases by 1.
  16. A train travels at 60 mph. After t hours, distance d = 60t. If d = 240 miles, how long did it travel? — 4 h. 60t = 240 → t = 240 ÷ 60 = 4 hours.
  17. A proportional relationship has constant ratio k = y/x = 7.5. When y = 45, x = ? — 6. y/x = 7.5 → x = y/7.5 = 45/7.5 = 6.
  18. Which graph shows a NON-proportional linear relationship? (hint: what does y-intercept ≠ 0 mean?) — y = 3x + 5 (y-intercept = 5). y = 3x + 5 has y-intercept = 5 (not the origin). It is linear but NOT proportional. Proportional must pass through (0,0).
  19. A pool fills at a rate of 120 gallons/hour. Starting from empty, after 3.5 hours it holds: — 420 gallons. 120 × 3.5 = 420 gallons.
  20. Two students save money. Student A saves $10/week. Student B saves $15/week. After w weeks, A has $10w and B has $15w. Difference after 8 weeks: — $40. After 8 weeks: A has $80, B has $120. Difference = $120 − $80 = $40.
  21. A machine produces 500 parts per hour. After 2.5 hours, how many parts? — 1,250. 500 × 2.5 = 1,250 parts.
  22. y = kx and the point (4, 10) is on the line. What is k? — 2.5. k = y/x = 10/4 = 2.5.
  23. Which statement about proportional relationships is INCORRECT? — The y-intercept can be any value. INCORRECT statement: y-intercept can be any value. For a PROPORTIONAL relationship, the y-intercept must be 0 (passes through origin). If y-intercept ≠ 0, it is non-proportional.