A bag has coins of 50 paisa, 25 paisa and 10 paisa in the respective ratio of 5 : 8 : 3 whose total value is ₹ 144. Find the number of 50 paisa coins.

### Concept & Value of Coins When given the ratio of the number of coins of different denominations and their total value, express the total value in terms of a common variable. $$ \text{Total Value} = \sum (\text{Number of Coins} \times \text{Denomination Value}) $$ ### Step-by-Step Solution * **Given:** Ratio of coins = 5 : 8 : 3. Total value = ₹ 144. Denominations = 50p (₹ 0.50), 25p (₹ 0.25), and 10p (₹ 0.10). * Let the number of 50 paisa, 25 paisa, and 10 paisa coins be $5x$, $8x$, and $3x$ respectively. * Calculate the value contributed by each type of coin in Rupees: * Value of 50p coins = $5x \times 0.50 = 2.5x$ * Value of 25p coins = $8x \times 0.25 = 2.0x$ * Value of 10p coins = $3x \times 0.10 = 0.3x$ * Sum these values to equal the total amount given: * $2.5x + 2.0x + 0.3x = 144$ * $4.8x = 144$ * Solve for $x$: * $x = \frac{144}{4.8} = \frac{1440}{48} = 30$ * We need the number of 50 paisa coins, which is $5x$: * Number of 50p coins = $5 \times 30 = 150$ ### Exam Strategy & Shortcut Work entirely in paise to avoid decimals. The values per $x$ proportion become $5 \times 50 = 250$, $8 \times 25 = 200$, and $3 \times 10 = 30$. Total parts = $250 + 200 + 30 = 480$. The total amount is $144 \times 100 = 14400$ paise. $1 \text{ part} = \frac{14400}{480} = 30$. Number of 50p coins = $5 \times 30 = 150$. ### Common Pitfall A frequent error is adding the ratios directly as if they represent the value (e.g., $5x + 8x + 3x = 144$). You must multiply the ratio by the respective denomination first. ### Final Answer Therefore, the correct answer is **150**.