A sum of ₹ 6.25 is made up of 80 coins which are either 10P or 5P. How many are there of each kind?

## Concept & Strategy This is a classic mixture problem that can be solved using simultaneous equations or the rule of alligation. The fastest approach is treating it as a weighted average problem (Alligation). ## Step-by-Step Solution * **Given:** Total number of coins = 80 Total value = ₹ 6.25 = 625 Paise Denominations = 10 Paise and 5 Paise * **Calculation (Using Equations):** Let the number of 10P coins be $x$. Then, the number of 5P coins will be $80 - x$. Set up the total value equation: $10x + 5(80 - x) = 625$ Expand and simplify: $10x + 400 - 5x = 625$ $5x + 400 = 625$ $5x = 225$ $x = 45$ So, there are 45 coins of 10P. Number of 5P coins = $80 - 45 = 35$. ## Exam Strategy & Shortcut Use the **Rule of Alligation** for lightning-fast solving. Assume ALL 80 coins are 10P $\implies 80 \times 10 = 800$ Paise. Assume ALL 80 coins are 5P $\implies 80 \times 5 = 400$ Paise. Actual mean value is 625 Paise. Apply alligation cross-subtraction: Ratio of 10P to 5P = $(625 - 400) : (800 - 625)$ $= 225 : 175$ Simplify by dividing by 25: $= 9 : 7$ Total parts = $9 + 7 = 16$. Number of 10P coins = $\frac{9}{16} \times 80 = 9 \times 5 = 45$. Number of 5P coins = $\frac{7}{16} \times 80 = 7 \times 5 = 35$. ## Common Pitfall Mixing up the final order in the answer options. Since the question asks for "10P or 5P" in that order, the answer must be listed as (10P count, 5P count), which is 45, 35. Choosing 35, 45 is a tragic unforced error. ## Final Answer **Therefore, the correct answer is 45, 35.**