A bag contains 25 paise coins, 50 paise coins, and 1 rupee coins. There are 220 coins in total, and the total value is ₹160. If the number of 1 rupee coins is three times the number of 25 paise coins, how many 50 paise coins are in the bag?

Introduction / Context:
This is a classic linear-equations word problem involving coin counts and total value. It tests forming equations from constraints (total coins, total money, and a ratio condition) and solving systematically.

Given Data / Assumptions:

• Let x = number of 25 paise coins • Let y = number of 50 paise coins • Let z = number of ₹1 coins • Total coins: x + y + z = 220 • Total value: 0.25x + 0.5y + 1z = 160 • Ratio: z = 3x

Concept / Approach:
Translate each statement into an equation. Use the ratio equation to substitute for z, reducing the system to two equations in two variables. Solve for y (the number of 50 paise coins).

Step-by-Step Solution:
1) From ratio: z = 3x 2) Substitute into total coins: x + y + 3x = 220 4x + y = 220 ...(Equation 1) 3) Substitute into total value: 0.25x + 0.5y + 3x = 160 4) Combine x terms: 0.25x + 3x = 3.25x 3.25x + 0.5y = 160 5) Multiply by 4 to clear decimals: 13x + 2y = 640 ...(Equation 2) 6) From (1): y = 220 − 4x 7) Substitute into (2): 13x + 2(220 − 4x) = 640 13x + 440 − 8x = 640 5x = 200 → x = 40 8) Then y = 220 − 4*40 = 60

Verification / Alternative check:
If x = 40, then z = 3x = 120. Total coins: 40 + 60 + 120 = 220. Total value: 40*0.25 + 60*0.5 + 120*1 = 10 + 30 + 120 = 160. Checks out.

Why Other Options Are Wrong:
• 40 or 80: would break either the total value equation or the total-coin equation after applying z = 3x. • 120 or 20: too large/too small to satisfy both constraints simultaneously.

Common Pitfalls:
• Treating 25 paise as 25 rupees instead of ₹0.25. • Forgetting to apply the ratio z = 3x.

Final Answer:
60