Equations with Fractions — Solutions

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1. Simple fraction equations

   1. x/2 = 5: x = 10
   2. x/3 + 1 = 4: x/3 = 3 → x = 9
   3. y/4 = −5: y = −20
   4. (a + 1)/2 = 6: a + 1 = 12 → a = 11
   5. (m − 3)/5 = 4: m − 3 = 20 → m = 23
   6. (2x + 1)/3 = 5: 2x + 1 = 15 → 2x = 14 → x = 7
   7. n/4 − 3 = 1: n/4 = 4 → n = 16
   8. (4k − 3)/5 = 3: 4k − 3 = 15 → 4k = 18 → k = 4.5

2. LCD method

   1. x/2 + x/3 = 10: LCD = 6. 3x + 2x = 60 → 5x = 60 → x = 12
   2. y/4 − y/6 = 2: LCD = 12. 3y − 2y = 24 → y = 24
   3. a/3 + 2 = a/2: LCD = 6. 2a + 12 = 3a → a = 12
   4. m/5 + m/4 = 9: LCD = 20. 4m + 5m = 180 → 9m = 180 → m = 20
   5. x/2 + 3 = x/4 + 7: LCD = 4. 2x + 12 = x + 28 → x = 16
   6. 2x/3 − x/4 = 5: LCD = 12. 8x − 3x = 60 → 5x = 60 → x = 12
   7. (x+1)/2 + (x+3)/3 = 5: LCD = 6. 3(x+1) + 2(x+3) = 30 → 3x+3+2x+6 = 30 → 5x = 21 → x = 4.2
   8. (2m−1)/3 − (m+2)/6 = 1: LCD = 6. 2(2m−1) − (m+2) = 6 → 4m−2−m−2 = 6 → 3m = 10 → m = 10/3

3. Binomial numerator equations

   1. (x+1)/2 = 3: x + 1 = 6 → x = 5
   2. (2x−3)/5 = 1: 2x − 3 = 5 → 2x = 8 → x = 4
   3. (x−2)/3 = (x+4)/5: Cross-multiply: 5(x−2) = 3(x+4) → 5x−10 = 3x+12 → 2x = 22 → x = 11
   4. (3x+1)/4 = (x+3)/2: LCD = 4: 3x+1 = 2(x+3) → 3x+1 = 2x+6 → x = 5
   5. (2x+3)/4 = (x−1)/2: LCD = 4: 2x+3 = 2(x−1) → 2x+3 = 2x−2 → 3 = −2. No solution (contradiction).
   6. (x+5)/3 = (2x−1)/4: Cross-multiply: 4(x+5) = 3(2x−1) → 4x+20 = 6x−3 → 23 = 2x → x = 11.5
   7. (4x−1)/3 = (2x+5)/2: Cross-multiply: 2(4x−1) = 3(2x+5) → 8x−2 = 6x+15 → 2x = 17 → x = 8.5
   8. (x−4)/6 = (x+2)/9: Cross-multiply: 9(x−4) = 6(x+2) → 9x−36 = 6x+12 → 3x = 48 → x = 16

4. True or False

   1. x = 10 in x/2 + x/3 = 10: 5 + 10/3 = 5 + 3.33 = 8.33 ≠ 10. False. Correct answer is x = 12.
   2. x = 5 in x/3 + x/6 = 5: 5/3 + 5/6 = 10/6 + 5/6 = 15/6 = 2.5 ≠ 5. False. Correct: LCD=6. 3x = 30 → x = 10.
   3. x = 11 in (x−2)/3 = (x+4)/5: (11−2)/3 = 9/3 = 3; (11+4)/5 = 15/5 = 3. 3 = 3 ✓ True
   4. x = 4.5 in (4x−3)/5 = 3: (4(4.5)−3)/5 = (18−3)/5 = 15/5 = 3 ✓ True
   5. x = 1/2 in 3/x = 6: 3 ÷ (1/2) = 6 ✓ True
   6. (2x+3)/4 = (x−1)/2 has no solution: LCD=4: 2x+3 = 2(x−1) → 2x+3 = 2x−2 → 3 = −2 (impossible). True — no solution.

5. Solve and verify

   1. 5/(x + 1) = 2:

      5 = 2(x + 1) → 5 = 2x + 2 → 2x = 3 → x = 1.5
      Check: 5 ÷ (1.5 + 1) = 5 ÷ 2.5 = 2 ✓

   2. 3/x = 6:

      3 = 6x → x = 0.5
      Check: 3 ÷ 0.5 = 6 ✓

6. Forming fraction equations

   1. x/3 + 5 = x/2:

      LCD = 6: 2x + 30 = 3x → x = 30

   2. x/4 + 2 = x/2 − 1:

      LCD = 4: x + 8 = 2x − 4 → 12 = x → x = 12

7. Savings (Aidan and Sophie)

   1. Equation: x/3 = x/4 + 10. LCD = 12: 4x = 3x + 120 → x = 120
   2. Each person saved x/3 = 120/3 = $40. Check: x/4 + 10 = 30 + 10 = 40 ✓

8. Spot the mistake

   1. Substitute x = 5: 5/3 + 5/6 = 10/6 + 5/6 = 15/6 = 2.5 ≠ 5. x = 5 is incorrect.
   2. The error: The student multiplied x/3 by 6 to get x (correct), but also multiplied x/6 by 6 to get x (correct), giving x + x = 5. However, they did not multiply the right-hand side (5) by 6. They should have got 2x + x ≠ 5 — actually the error is they wrote x + x when it should be LCD × x/3 + LCD × x/6 = 2x + x = 3x, and the right side = 6 × 5 = 30.
   3. Correct solution: LCD = 6. 2x + x = 30 → 3x = 30 → x = 10. Check: 10/3 + 10/6 = 20/6 + 10/6 = 30/6 = 5 ✓

9. Pipes filling a tank (Problem Solving)

   1. Equation: t/3 + t/4 = 1
   2. Solve: LCD = 12. 4t + 3t = 12 → 7t = 12 → t = 12/7
   3. In hours and minutes: 12/7 ≈ 1.714 hours = 1 hour 43 minutes

10. Scaled recipe (Problem Solving)

    1. Equation: (3/4)b = 9
    2. Solve: b = 9 × (4/3) = 36/3 = 12 cups
    3. How many times larger: The standard batch uses 1 cup. The scaled batch is 12 cups. It is 12 times larger.