There are 420 coins consisting of one-rupee coins, 50-paise coins and 25-paise coins. If the ratio of their values be 2 : 3 : 5, then the number of one-rupee coins is

### Concept & Coins from Value Ratio When given the ratio of the *values* of different coin denominations, you must divide each value part by the respective denomination's value to find the ratio of the *number* of coins. ### Step-by-Step Solution 1. The ratio of the values of one-rupee, 50-paise, and 25-paise coins is given as $2 : 3 : 5$. 2. Let their total values be $2x$, $3x$, and $5x$ in Rupees. 3. Calculate the number of coins for each denomination by dividing the value by the denomination value in Rupees: - Number of ₹1 coins = $\frac{2x}{1} = 2x$ - Number of 50p (₹0.50) coins = $\frac{3x}{0.50} = 6x$ - Number of 25p (₹0.25) coins = $\frac{5x}{0.25} = 20x$ 4. The total number of coins is given as 420. Sum the number of coins and equate: $$ 2x + 6x + 20x = 420 $$ $$ 28x = 420 $$ 5. Solve for $x$: $$ x = \frac{420}{28} = 15 $$ 6. The question asks for the number of one-rupee coins, which we defined as $2x$. 7. Number of one-rupee coins = $2(15) = 30$. ### Exam Strategy & Shortcut Work directly with the ratios without variables. Value ratio = $2 : 3 : 5$. To get the number of coins, multiply the value by how many of those coins make one Rupee: ₹1 coins (1 per Rupee) $\rightarrow 2 \times 1 = 2$ 50p coins (2 per Rupee) $\rightarrow 3 \times 2 = 6$ 25p coins (4 per Rupee) $\rightarrow 5 \times 4 = 20$ The ratio of the number of coins is $2 : 6 : 20$, which simplifies to $1 : 3 : 10$. Total ratio parts = $1 + 3 + 10 = 14$ parts. 14 parts = 420 coins, meaning 1 part = 30 coins. The number of ₹1 coins is 1 part, which equals 30. ### Common Pitfall A frequent error is assuming the given ratio of values ($2:3:5$) represents the ratio of the number of coins, and mistakenly calculating $2x + 3x + 5x = 420$. ### Final Answer Therefore, the correct answer is **30**.