Understanding the Coin Problem and Ratio

The question asks us to find the number of one rupee, 50 paisa, and 25 paisa coins given their total value and the ratio of their numbers. We are told that the total value of the coins is ₹93.75 and the number of one rupee, 50 paisa, and 25 paisa coins are in the ratio of 3 : 4 : 5.

Let's break down the given information:

• Coin denominations: ₹1, 50 paisa, 25 paisa.
• Total value: ₹93.75
• Ratio of the number of coins (₹1 : 50 paisa : 25 paisa): 3 : 4 : 5

Setting up the Equation for Total Value

To solve this, we can represent the number of coins using the given ratio and a variable. Let the common ratio factor be \(x\).

• Number of ₹1 coins = \(3x\)
• Number of 50 paisa coins = \(4x\)
• Number of 25 paisa coins = \(5x\)

Next, we need to express the value of each type of coin in a common unit. Paisa is easier for calculations involving 50 and 25 paisa coins. We know that 1 rupee = 100 paisa.

• Value of ₹1 coin = 100 paisa
• Value of 50 paisa coin = 50 paisa
• Value of 25 paisa coin = 25 paisa

The total value of the coins in paisa is ₹93.75. We convert this to paisa:

\(93.75 \text{ rupees} = 93.75 \times 100 \text{ paisa} = 9375 \text{ paisa}\)

Now we can set up an equation based on the total value. The total value is the sum of the values of all the coins.

Value of \(3x\) coins of ₹1 = \(3x \times 100\) paisa

Value of \(4x\) coins of 50 paisa = \(4x \times 50\) paisa

Value of \(5x\) coins of 25 paisa = \(5x \times 25\) paisa

The total value equation is:

\((3x \times 100) + (4x \times 50) + (5x \times 25) = 9375\)

\(300x + 200x + 125x = 9375\)

Solving for the Unknown Variable \(x\)

Now we simplify and solve the linear equation for \(x\).

Combine the terms with \(x\):

\((300 + 200 + 125)x = 9375\)

\(625x = 9375\)

To find \(x\), we divide the total value by the sum of the weighted ratios:

\(x = \frac{9375}{625}\)

Performing the division:

\(9375 \div 625 = 15\)

So, the value of \(x\) is 15.

Calculating the Number of Each Type of Coins

Now that we have the value of \(x\), we can find the number of each type of coin using the expressions we defined earlier:

• Number of ₹1 coins = \(3x = 3 \times 15 = 45\)
• Number of 50 paisa coins = \(4x = 4 \times 15 = 60\)
• Number of 25 paisa coins = \(5x = 5 \times 15 = 75\)

Thus, the number of one rupee, 50 paisa, and 25 paisa coins are 45, 60, and 75 respectively.

Verifying the Solution

Let's check if these numbers give the correct total value.

• Value of 45 coins of ₹1 = \(45 \times 1\) rupee = ₹45
• Value of 60 coins of 50 paisa = \(60 \times 0.50\) rupees = ₹30
• Value of 75 coins of 25 paisa = \(75 \times 0.25\) rupees = ₹18.75

Total value = ₹45 + ₹30 + ₹18.75 = ₹93.75

This matches the total value given in the question, confirming our calculated numbers are correct.

The number of coins are 45, 60, and 75.

Revision Table: Coin Problem Summary

Total value in paisa = 9375

Equation: \(625x = 9375\)

Solution: \(x = 15\)

Number of ₹1 coins = \(3 \times 15 = 45\)

Number of 50 paisa coins = \(4 \times 15 = 60\)

Number of 25 paisa coins = \(5 \times 15 = 75\)

Additional Information on Ratio and Proportion

A ratio is a way to compare two or more quantities. In this problem, the ratio 3 : 4 : 5 tells us that for every 3 one rupee coins, there are 4 fifty paisa coins and 5 twenty-five paisa coins. Proportion is an equation that states that two ratios are equal.

When dealing with ratios involving different units (like different coin values), it's crucial to convert everything to a single unit before setting up equations related to total value. In this case, converting rupees to paisa or paisa to rupees is necessary.

Using a variable like \(x\) with the ratio (e.g., \(3x, 4x, 5x\)) is a standard technique to represent quantities that are in a given ratio, allowing us to set up and solve algebraic equations.