I'll solve each of these linear equation word problems step by step.
Problem 1: Sarah's delivery charges
- Flat fee: $20
- Per kilometer: $5
- Goal: at least $150
- Let d = distance in kilometers
- Equation: 20 + 5d ≥ 150
- 5d ≥ 130
- d ≥ 26
Answer: At least 26 kilometers
---
Problem 2: Coffee blend
- Brazilian beans: $8/pound
- Colombian beans: $12/pound
- Blend sells for: $10/pound
- Total blend: 50 pounds
- Let b = pounds of Brazilian beans
- Then (50-b) = pounds of Colombian beans
- Cost equation: 8b + 12(50-b) = 10(50)
- 8b + 600 - 12b = 500
- -4b = -100
- b = 25
Answer: 25 pounds of Brazilian beans, 25 pounds of Colombian beans
---
Problem 3: Rectangular garden
- Length: 5 meters longer than width
- Area: 60 square meters
- Let w = width
- Then length = w + 5
- Area equation: w(w + 5) = 60
- w² + 5w = 60
- w² + 5w - 60 = 0
- (w + 12)(w - 5) = 0
- w = 5 (width can't be negative)
- Length = 5 + 5 = 10
Answer: Width = 5 meters, Length = 10 meters
---
Problem 4: Savings account
- Initial deposit: $5000
- Annual interest: 3%
- Monthly deposit: $100
- Goal: $10,000
- This is complex with compound interest, but using simple approach:
- After n years: 5000(1.03)^n + 100×12×n ≈ 10000
- Testing: Year 3: 5000(1.093) + 3600 = 5465 + 3600 = 9065
- Year 4: 5000(1.126) + 4800 = 5630 + 4800 = 10430
Answer: About 4 years
---
Problem 5: Train meeting time
- Distance between A and B: 300 miles
- Train from A: 60 mph toward B
- Train from B: 40 mph toward A
- Combined speed: 60 + 40 = 100 mph
- Time to meet: 300 ÷ 100 = 3 hours
Answer: 3 hours
---
Problem 6: Phone plans
- Plan A: $30/month + $0.05/minute
- Plan B: $50/month (unlimited)
- Let m = minutes
- Plan A cost: 30 + 0.05m
- Plan B cheaper when: 50 < 30 + 0.05m
- 20 < 0.05m
- 400 < m
Answer: More than 400 minutes per month
---
Problem 7: Chair production
- Cost per chair: $20
- Setup fee: $500
- Selling price: $50/chair
- Goal profit: $2000
- Let c = number of chairs
- Revenue: 50c
- Cost: 20c + 500
- Profit: 50c - (20c + 500) = 2000
- 30c - 500 = 2000
- 30c = 2500
- c = 83.33
Answer: 84 chairs (must round up)
---
Problem 8: Daily steps
- Current: 5000 steps/day
- Increase: 500 steps/day each week
- Goal: 10,000 steps/day
- Additional needed: 10000 - 5000 = 5000 steps
- Weeks needed: 5000 ÷ 500 = 10 weeks
Answer: 10 weeks
---
Problem 9: Shower water savings
- Old flow: 2.5 gallons/minute
- New flow: 1.5 gallons/minute
- Savings per minute: 2.5 - 1.5 = 1 gallon/minute
- Shower time: 15 minutes
- Daily savings: 1 × 15 = 15 gallons
- Weekly savings: 15 × 7 = 105 gallons
Answer: 105 gallons per week
---
Problem 10: Construction crew
- Full crew (6 workers): 12 hours
- One worker alone: would take 6 × 12 = 72 hours
- Sick worker does same work as one regular worker
Answer: 72 hours
Parent Tip: Review the logic above to help your child master the concept of linear system word problems worksheet.
