Coins, Income & Mixture Ratios

🎯 Learning Objective

Solve coin-bag value questions and income-expenditure-savings ratio problems.

💡 Concept

• Coin bags: the ratio is of the NUMBER of coins, but the total is in VALUE (₹)
• Convert one ratio-set to value: for ₹1, 50p, 25p in a:b:c → set value = a(1) + b(0.5) + c(0.25)
• Number of sets = total value ÷ set value; then coins of each type = ratio term × sets
• Income − Expenditure = Savings; put incomes as 4x, 5x and expenditures as 5y, 7y
• Two equations from two savings → solve x and y

🧮 Key Formulas

Set value = a×1 + b×0.50 + c×0.25

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Sets = Total value/Set value

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Income − Expenditure = Savings

✏️ Easy Example

Q. A bag has ₹1, 50 p and 25 p coins in the ratio 2:3:4, worth ₹90 in total. Find the number of ₹1 coins.

1. Set value = 2×1 + 3×0.5 + 4×0.25 = ₹4.5
2. Sets = 90/4.5 = 20
3. ₹1 coins = 2 × 20

Answer: 40 coins

🇮🇳 Real-Life Example

A bus conductor's pouch holds coins in a fixed mix — he counts the value, not each coin. Set-value maths is exactly how he tallies at night.

📝 Exam-Level Example

Q. A bag contains 50 p, 25 p and 10 p coins in the ratio 5:9:4, amounting to ₹206. Find the number of 25 p coins.

1. Set value = 5×0.50 + 9×0.25 + 4×0.10 = ₹5.15
2. Sets = 206/5.15 = 40
3. 25 p coins = 9 × 40

Answer: 360 coins

📝 Exam-Level Example

Q. Incomes of A and B are in the ratio 4:5 and their expenditures in the ratio 5:7. If each saves ₹3,000, find their incomes.

1. 4x − 5y = 3000 and 5x − 7y = 3000
2. Subtract: x − 2y = 0 → x = 2y
3. 8y − 5y = 3000 → y = 1000, x = 2000
4. Incomes = 4x and 5x

Answer: ₹8,000 and ₹10,000

🪄 Memory Trick

Coins: never work coin-by-coin — price ONE ratio-set, divide the total by it. Incomes: two savings lines give two equations, subtract them first.

⚠️ Common Mistakes

• ❌ Treating the coin ratio as a ratio of values instead of counts
• ❌ Forgetting to convert paise to rupees (50 p = 0.50, not 50)
• ❌ Assuming equal savings means equal incomes

🏆 Exam Tips

• ✅ Keep everything in rupees from step one — paise mixing causes 90% of errors
• ✅ In income questions, savings equations subtract cleanly when savings are equal

📌 Summary

• Coin ratio = counts; total = value
• Set value → sets → coins of each type
• Income − Expenditure = Savings, always
• Two savings facts → two equations → solve