Let’s solve each equation one by one. We’ll isolate the variable (usually x) using inverse operations — that means if something is added, we subtract; if something is multiplied, we divide, and so on.
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① (2/3)x - 15 = 65
Step 1: Add 15 to both sides
→ (2/3)x = 80
Step 2: Multiply both sides by the reciprocal of 2/3, which is 3/2
→ x = 80 × (3/2) = (80 × 3)/2 = 240/2 =
120
✔ Check: (2/3)(120) - 15 = 80 - 15 = 65 ✔️
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② 2x = 49/5
Step 1: Divide both sides by 2 (or multiply by 1/2)
→ x = (49/5) × (1/2) = 49/10
✔ Final answer:
49/10 or 4.9
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③ (9/10)x = -11/10
Step 1: Multiply both sides by reciprocal of 9/10 → 10/9
→ x = (-11/10) × (10/9) = -110/90 = simplify by dividing numerator and denominator by 10 →
-11/9
✔ Check: (9/10)(-11/9) = -99/90 = -11/10 ✔️
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④ 12/5 = 1/3 + x
Step 1: Subtract 1/3 from both sides
→ x = 12/5 - 1/3
Find common denominator: 15
→ 12/5 = 36/15, 1/3 = 5/15
→ x = 36/15 - 5/15 = 31/15
✔ Final answer:
31/15
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⑤ x - 4/7 = 14
Step 1: Add 4/7 to both sides
→ x = 14 + 4/7 = 98/7 + 4/7 = 102/7
✔ Final answer:
102/7
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⑥ x - (x - 1)/2 = 0
Step 1: Get rid of fraction by multiplying entire equation by 2
→ 2x - (x - 1) = 0
Step 2: Distribute the negative sign
→ 2x - x + 1 = 0
→ x + 1 = 0
Step 3: Subtract 1
→ x = -1
✔ Check: -1 - [(-1 -1)/2] = -1 - [-2/2] = -1 - (-1) = -1 + 1 = 0 ✔️
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⑦ 1/3 = x + 4/3
Step 1: Subtract 4/3 from both sides
→ x = 1/3 - 4/3 = -3/3 =
-1
✔ Check: -1 + 4/3 = 1/3 ✔️
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⑧ 1/2 + x/3 = x/2
Step 1: Get all x terms on one side. Subtract x/3 from both sides
→ 1/2 = x/2 - x/3
Step 2: Find common denominator for right side (6)
→ x/2 = 3x/6, x/3 = 2x/6
→ 1/2 = 3x/6 - 2x/6 = x/6
Step 3: Multiply both sides by 6
→ x = 6 × (1/2) = 3
✔ Check: 1/2 + 3/3 = 1/2 + 1 = 3/2; and x/2 = 3/2 ✔️
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⑨ x - 3/9 = 15
Simplify 3/9 → 1/3
→ x - 1/3 = 15
Add 1/3 to both sides
→ x = 15 + 1/3 = 45/3 + 1/3 = 46/3
✔ Final answer:
46/3
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⑩ (2x - 1)/3 + 3 = x
Step 1: Subtract 3 from both sides
→ (2x - 1)/3 = x - 3
Step 2: Multiply both sides by 3
→ 2x - 1 = 3(x - 3)
Step 3: Expand right side
→ 2x - 1 = 3x - 9
Step 4: Subtract 2x from both sides
→ -1 = x - 9
Step 5: Add 9 to both sides
→ x = 8
✔ Check: (2*8 -1)/3 + 3 = (16-1)/3 + 3 = 15/3 + 3 = 5 + 3 = 8 ✔️
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Final Answer:
① 120
② 49/10
③ -11/9
④ 31/15
⑤ 102/7
⑥ -1
⑦ -1
⑧ 3
⑨ 46/3
⑩ 8
Parent Tip: Review the logic above to help your child master the concept of equations with rational numbers worksheet.
