Word Problems
Overview
Learn to translate real-world situations into systems of linear equations. This lesson focuses on setting up and solving word problems involving two unknowns.
Practice Problems
Question 1: The sum of two numbers is 25 and their difference is 7. Find both numbers.
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Answer: 16 and 9
Let x + y = 25 and x - y = 7. Adding: 2x = 32, x = 16. Then y = 25 - 16 = 9.
Question 2: Movie tickets cost $12 for adults and $8 for children. A group of 10 people paid $104 total. How many adults and children were there?
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Answer: 6 adults and 4 children
Let a + c = 10 and 12a + 8c = 104. From first: c = 10 - a. Substitute: 12a + 8(10-a) = 104, 4a = 24, a = 6, c = 4.
Question 3: A rectangle's perimeter is 36 cm. Its length is 4 cm more than its width. Find the dimensions.
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Answer: Length = 11 cm, Width = 7 cm
2L + 2W = 36 and L = W + 4. Substitute: 2(W+4) + 2W = 36, 4W = 28, W = 7, L = 11.
Question 4: Maria has $3.50 in quarters and dimes. She has 20 coins in total. How many of each coin does she have?
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Answer: 10 quarters and 10 dimes
q + d = 20 and 0.25q + 0.10d = 3.50. From first: d = 20 - q. Substitute: 0.25q + 0.10(20-q) = 3.50, 0.15q = 1.50, q = 10, d = 10.
Question 5: A boat travels 30 miles downstream in 2 hours and 18 miles upstream in 3 hours. Find the speed of the boat in still water and the speed of the current.
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Answer: Boat speed = 10.5 mph, Current speed = 4.5 mph
Downstream: b + c = 15. Upstream: b - c = 6. Adding: 2b = 21, b = 10.5, c = 4.5.
Question 6: Two angles are supplementary. One angle is 30 degrees more than twice the other. Find both angles.
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Answer: 50 degrees and 130 degrees
x + y = 180 and y = 2x + 30. Substitute: x + 2x + 30 = 180, 3x = 150, x = 50, y = 130.
Question 7: A store sold 85 shirts and pants. Shirts cost $25 and pants cost $40. Total sales were $2,600. How many of each were sold?
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Answer: 60 shirts and 25 pants
s + p = 85 and 25s + 40p = 2600. Substitute s = 85 - p: 25(85-p) + 40p = 2600, 15p = 475, p = 25, s = 60.
Question 8: Jen is 4 years older than her brother. In 5 years, the sum of their ages will be 36. How old is each now?
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Answer: Jen is 15, her brother is 11
J = B + 4 and (J+5) + (B+5) = 36. Substitute: (B+4+5) + (B+5) = 36, 2B = 22, B = 11, J = 15.
Question 9: A mixture contains nuts and raisins. Nuts cost $6/lb and raisins cost $4/lb. If 10 pounds of the mixture costs $48, how many pounds of each are in the mixture?
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Answer: 4 pounds of nuts and 6 pounds of raisins
n + r = 10 and 6n + 4r = 48. Substitute r = 10 - n: 6n + 4(10-n) = 48, 2n = 8, n = 4, r = 6.
Question 10: A test has 30 questions worth 100 points. Some questions are worth 3 points and others are worth 4 points. How many of each type are on the test?
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Answer: 20 three-point questions and 10 four-point questions
a + b = 30 and 3a + 4b = 100. Substitute a = 30 - b: 3(30-b) + 4b = 100, b = 10, a = 20.