Coins Ratio Problem Explained

This question asks us to determine the quantity of ₹5 coins within a bag, given specific information about the number ratio of different coin denominations (₹5, ₹2, and ₹1) and their total combined value.

Setting Up the Coin Ratio

The bag contains ₹5, ₹2, and ₹1 coins.

The ratio of the number of these coins is provided as 3:4:5.

To work with this ratio, let's introduce a common multiplier, '$k$'. This allows us to represent the number of coins for each denomination:

• Number of ₹5 coins = $3k$
• Number of ₹2 coins = $4k$
• Number of ₹1 coins = $5k$

Calculating Total Value from Denominations

The total value of all coins is ₹2,100. We can calculate the value contributed by each set of coins:

• The value contributed by the ₹5 coins is: (Number of ₹5 coins) $\times$ ₹5 = $3k \times 5 = 15k$
• The value contributed by the ₹2 coins is: (Number of ₹2 coins) $\times$ ₹2 = $4k \times 2 = 8k$
• The value contributed by the ₹1 coins is: (Number of ₹1 coins) $\times$ ₹1 = $5k \times 1 = 5k$

Formulating the Total Value Equation

The sum of the values from each denomination equals the total value of the coins in the bag:

$$ \text{Total Value} = (15k) + (8k) + (5k) $$

We know the total value is ₹2,100, so we can write the equation as:

$$ 2100 = 15k + 8k + 5k $$

Solving for the Unknown Multiplier

Combine the terms involving '$k$' on the right side of the equation:

$$ 2100 = (15 + 8 + 5)k $$

$$ 2100 = 28k $$

Now, isolate '$k$' by dividing both sides by 28:

$$ k = \frac{2100}{28} $$

To simplify the calculation, we can divide both the numerator and the denominator by common factors. Dividing by 7:

$$ k = \frac{2100 \div 7}{28 \div 7} = \frac{300}{4} $$

Performing the division:

$$ k = 75 $$

Thus, the common multiplier '$k$' is 75.

Finding the Number of ₹5 Coins

The question specifically asks for the number of ₹5 coins.

We established earlier that the number of ₹5 coins is represented by $3k$.

Substitute the value of '$k = 75$' into this expression:

Number of ₹5 coins = $3 \times 75$

$$ \text{Number of } ₹5 \text{ coins} = 225 $$

This step-by-step method, based on the provided ratio and total value, results in 225 coins of ₹5.