Equations and inequalities · Ages 13–14

Word problems with simultaneous equations worksheet

Part of: Simultaneous linear equations

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Equations and inequalities · Ages 13–14

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.1 adult and 1 child ticket cost $110; 2 adult and 3 child tickets cost $270. An adult ticket is $ and a child's $.
Fill in each blank
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2.Chickens and rabbits: 21 heads and 56 legs. How many rabbits are there?
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3.1 adult and 1 child ticket cost $120; 1 adult and 3 child tickets cost $180. An adult ticket is $ and a child's $.
Fill in each blank
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4.Chickens and rabbits: 22 heads and 58 legs. How many rabbits are there?
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5.$5 and $2 coins, 24 in all, worth $87. There are $5 coins and $2 coins.
Fill in each blank
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6.A two-digit number's digits add to 8. Swapping its tens and ones digits gives a number 36 larger. What is the number?
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7.Two numbers add to 59 and differ by 13. The larger is and the smaller .
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8.Two numbers add to 64 and differ by 40. The larger is and the smaller .
Fill in each blank
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9.A boat goes 36 km downstream in 3 hours and the same way back upstream in 18 hours. Its speed in still water is km/h and the current's km/h.
Fill in each blank
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10.A father is 3 times as old as his son. In 13 years he will be 2 times as old. How old is the son now?
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11.A boat goes 48 km downstream in 4 hours and the same way back upstream in 6 hours. Its speed in still water is km/h and the current's km/h.
Fill in each blank
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12.A father is 5 times as old as his son. In 6 years he will be 3 times as old. How old is the son now?
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13.A boat takes 4 hours to go 48 km downstream and 12 hours to come back upstream. How fast is the current, in km/h?
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14.3 years ago, a mother was 2 times as old as her daughter; now their ages add to 24. How old is the mother now?
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15.4 adult and 1 child tickets cost $400; 2 adult and 3 child tickets cost $300. An adult ticket is $ and a child's $.
Fill in each blank
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16.A two-digit number's digits add to 9, and the number is 2 times its digit sum. What is the number?
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17.A cash box has $5 and $2 coins, 18 in all, and the $5 coins are worth $69 more than the $2 coins. How many $5 coins are there?
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18.Chickens and rabbits: 20 heads, 56 legs. With x chickens and y rabbits: x + y = 20 and 2x + 4y = 56. Put the steps of finding the rabbits in order.
- A. 2x + 2y = 40
- B. y = 8
- C. (2x + 4y) − (2x + 2y) = 56 − 40
- D. 2y = 16
Write the letters in the right order
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Show the answers and working

Word problems with simultaneous equations worksheet: Answers and working

1. 60, 50
x + y = 110, 2x + 3y = 270. Take 2 times the first from the second: y = 50, x = 60.
2. 7 rabbits
x + y = 21, 2x + 4y = 56. Or suppose all chickens: 21 × 2 legs; each rabbit adds 2, so rabbits = (56 − 2 × 21) ÷ 2 = 7.
3. 90, 30
x + y = 120, x + 3y = 180. Take 1 times the first from the second: y = 30, x = 90.
4. 7 rabbits
x + y = 22, 2x + 4y = 58. Or suppose all chickens: 22 × 2 legs; each rabbit adds 2, so rabbits = (58 − 2 × 22) ÷ 2 = 7.
5. 13, 11
x + y = 24, 5x + 2y = 87. Take twice the first from the second: 3x = 87 − 2 × 24, x = 13, y = 11.
6. 26
Tens digit x, ones digit y: x + y = 8, (10y + x) − (10x + y) = 36, so y − x = 4. Then x = 2, y = 6: the number is 26.
7. 36, 23
x + y = 59, x − y = 13. Adding: 2x = 59 + 13, x = 36; y = 59 − 36 = 23.
8. 52, 12
x + y = 64, x − y = 40. Adding: 2x = 64 + 40, x = 52; y = 64 − 52 = 12.
9. 7, 5
Downstream u + v = 36 ÷ 3, upstream u − v = 36 ÷ 18. Add and halve: u = 7; subtract and halve: v = 5.
10. 13 years old
Father x, son y: x = 3y and x + 13 = 2(y + 13). Substituting, 3y + 13 = 2y + 2 × 13, so y = 13.
11. 10, 2
Downstream u + v = 48 ÷ 4, upstream u − v = 48 ÷ 6. Add and halve: u = 10; subtract and halve: v = 2.
12. 6 years old
Father x, son y: x = 5y and x + 6 = 3(y + 6). Substituting, 5y + 6 = 3y + 3 × 6, so y = 6.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 4 km/h
u + v = 48 ÷ 4 and u − v = 48 ÷ 12. Subtracting: 2v = (48 ÷ 4) − (48 ÷ 12), v = 4.
14. 15 years old
x + y = 24 and x − 3 = 2(y − 3). With x = 24 − y: 24 − y − 3 = 2y − 2 × 3, so y = 9 and x = 15.
15. 90, 40
4x + y = 400 and 2x + 3y = 300. Eliminate one unknown: x = 90, y = 40. Check in both.
16. 18
x + y = 9 and 10x + y = 2(x + y) = 2 × 9. Subtracting: 9x = 2 × 9 − 9, x = 1, y = 8: the number is 18.
17. 15 coins
x + y = 18 and 5x − 2y = 69. Double the first and add: 7x = 69 + 2 × 18, x = 15.
18. A → C → D → B
Double the first, take it from the second to remove x, then halve.

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