Atul purchased Bread costing Rs.20 and gave a 100 rupee note to the shopkeeper. The shopkeeper gave the balance money in coins of denomination Rs.2, Rs.5 and Rs.10. If these coins are in the ratio 5 ∶ 4 ∶ 1, then how many Rs.5 coins did the shopkeeper give?
8
Understanding the Money Problem
This question asks us to determine the number of Rs. 5 coins a shopkeeper gave back as change. We are given the cost of an item, the amount paid, and the ratio of different coin denominations used for the change.
Calculating the Total Change Amount
First, let's find out the total amount of change Atul received from the shopkeeper. The change is the difference between the money paid and the cost of the bread.
Money paid = Rs. 100
Cost of Bread = Rs. 20
Total Change = Money paid - Cost of Bread
Total Change = \$100 - \$20 = \$80$
So, the shopkeeper gave back Rs. 80 in coins.
Setting Up the Coin Ratio and Value
The shopkeeper gave the change in coins of denominations Rs. 2, Rs. 5, and Rs. 10. The ratio of these coins is given as 5 : 4 : 1.
Let's represent the number of coins using a variable \(x\):
- Number of Rs. 2 coins = \$5x$
- Number of Rs. 5 coins = \$4x$
- Number of Rs. 10 coins = \$1x$ (or just \$x$)
Now, let's calculate the total value contributed by each type of coin based on this ratio:
- Value from Rs. 2 coins = (Number of Rs. 2 coins) \(\times\) (Denomination) = \$5x \times 2 = 10x$ rupees
- Value from Rs. 5 coins = (Number of Rs. 5 coins) \(\times\) (Denomination) = \$4x \times 5 = 20x$ rupees
- Value from Rs. 10 coins = (Number of Rs. 10 coins) \(\times\) (Denomination) = \$1x \times 10 = 10x$ rupees
Forming an Equation and Solving for x
The sum of the values from all coin denominations must equal the total change received, which is Rs. 80.
Total Value of Change = Value from Rs. 2 coins + Value from Rs. 5 coins + Value from Rs. 10 coins
\$80 = 10x + 20x + 10x$
Combine the terms on the right side:
\$80 = (10 + 20 + 10)x$
\$80 = 40x$
Now, solve for \(x\) by dividing both sides by 40:
\$x = \frac{80}{40}$
\$x = 2$
Finding the Number of Rs. 5 Coins
The question asks for the number of Rs. 5 coins. Based on our ratio setup, the number of Rs. 5 coins is \(4x\).
Number of Rs. 5 coins = \$4x$
Substitute the value of \(x\) we found:
Number of Rs. 5 coins = \$4 \times 2$
Number of Rs. 5 coins = \$8$
Therefore, the shopkeeper gave 8 coins of Rs. 5 denomination.
Summary of Coin Calculation
Let's quickly verify the total change value with \(x=2\):
- Number of Rs. 2 coins = \$5x = 5 \times 2 = 10$. Value = \$10 \times 2 = 20$
- Number of Rs. 5 coins = \$4x = 4 \times 2 = 8$. Value = \$8 \times 5 = 40$
- Number of Rs. 10 coins = \$1x = 1 \times 2 = 2$. Value = \$2 \times 10 = 20$
Total value = \$20 + 40 + 20 = 80$ rupees, which matches the required change amount.
The number of Rs. 5 coins is indeed 8.
Revision Table: Coin Denomination Ratio Problem
| Item | Value |
|---|---|
| Cost of Bread | Rs. 20 |
| Money Paid | Rs. 100 |
| Total Change | Rs. 80 |
| Ratio Rs. 2 : Rs. 5 : Rs. 10 | 5 : 4 : 1 |
| Let the multiplier be \(x\) | |
| Number of Rs. 2 coins | \$5x$ |
| Number of Rs. 5 coins | \$4x$ |
| Number of Rs. 10 coins | \$x$ |
| Equation for Total Value | \$2(5x) + 5(4x) + 10(x) = 80$ |
| Simplified Equation | \$10x + 20x + 10x = 80$ |
| \$40x = 80$ | |
| Value of \(x\) | \$x = 2$ |
| Number of Rs. 5 coins | \$4x = 4 \times 2 = 8$ |
Additional Information: Ratio and Proportion in Money Problems
Ratio and proportion are common concepts in quantitative aptitude problems, especially those involving division of amounts or objects based on a given ratio, like this coin denomination problem. A ratio represents a relative relationship between quantities. For example, a ratio of 5:4:1 means that for every 5 coins of the first type, there are 4 of the second, and 1 of the third.
When solving ratio problems, it's often helpful to introduce a variable, like \(x\), to represent the common multiplier for the ratio components. This allows you to convert the relative proportions into actual quantities (in terms of \(x\)) and then set up equations based on the total amount or value provided in the problem. Once \(x\) is found, you can calculate the specific quantity for any part of the ratio.
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