A bag contains Rs. 550 in the form of 50 p, 25 p and 20 p coins in the ratio 2 ∶ 3 ∶ 5. The difference between the amounts that are contributed by the 50 p and the 20 p coins is:
Rs. 0
Understanding the Problem: Coins in a Bag
The question asks us to find the difference in the total value contributed by the 50 paise (p) coins and the 20 paise coins in a bag. We are given the total amount of money in the bag (Rs. 550) and the ratio of the number of 50 p, 25 p, and 20 p coins (2:3:5).
Setting up the Coin Quantities
Let the number of 50 p, 25 p, and 20 p coins be represented by \(2x\), \(3x\), and \(5x\), respectively, based on the given ratio 2:3:5. Here, \(x\) is a common multiplier for the ratio.
Calculating the Value of Each Denomination
To find the total value, we need to convert the coin values to rupees. Remember:
- 1 Rupee = 100 paise
- 50 p = Rs. \(\frac{50}{100} = \) Rs. 0.50
- 25 p = Rs. \(\frac{25}{100} = \) Rs. 0.25
- 20 p = Rs. \(\frac{20}{100} = \) Rs. 0.20
Now, we can express the total value contributed by each type of coin in terms of \(x\):
- Value from 50 p coins = (Number of 50 p coins) \(\times\) (Value of one 50 p coin) = \(2x \times 0.50 = 1.00x\) Rupees
- Value from 25 p coins = (Number of 25 p coins) \(\times\) (Value of one 25 p coin) = \(3x \times 0.25 = 0.75x\) Rupees
- Value from 20 p coins = (Number of 20 p coins) \(\times\) (Value of one 20 p coin) = \(5x \times 0.20 = 1.00x\) Rupees
Forming and Solving the Total Value Equation
The total amount in the bag is given as Rs. 550. This total is the sum of the values contributed by all the coins:
Total Value = (Value from 50 p coins) + (Value from 25 p coins) + (Value from 20 p coins)
So, we have the equation:
\(1.00x + 0.75x + 1.00x = 550\)
Combine the terms with \(x\):
\((1.00 + 0.75 + 1.00)x = 550\)
\(2.75x = 550\)
To find the value of \(x\), divide both sides by 2.75:
\(x = \frac{550}{2.75}\)
We can write 2.75 as \(\frac{275}{100} = \frac{11}{4}\). So,
\(x = \frac{550}{\frac{11}{4}} = 550 \times \frac{4}{11}\)
\(x = \frac{550 \times 4}{11} = \frac{2200}{11}\)
\(x = 200\)
Calculating the Amount from Each Type of Coin
Now that we have the value of \(x\), we can find the exact amount contributed by each type of coin:
- Amount from 50 p coins = \(1.00x = 1.00 \times 200 = 200\) Rupees
- Amount from 25 p coins = \(0.75x = 0.75 \times 200 = 150\) Rupees
- Amount from 20 p coins = \(1.00x = 1.00 \times 200 = 200\) Rupees
Calculating the Difference in Amounts
The question asks for the difference between the amounts contributed by the 50 p and the 20 p coins.
Difference = (Amount from 50 p coins) - (Amount from 20 p coins)
Difference = Rs. \(200 - \) Rs. \(200 = \) Rs. \(0\)
Let's summarize the coin counts and values in a table:
| Coin Denomination | Ratio Part | Number of Coins (\(x=200\)) | Value per Coin (Rs.) | Total Amount (Rs.) |
|---|---|---|---|---|
| 50 p | 2 | \(2 \times 200 = 400\) | 0.50 | \(400 \times 0.50 = 200\) |
| 25 p | 3 | \(3 \times 200 = 600\) | 0.25 | \(600 \times 0.25 = 150\) |
| 20 p | 5 | \(5 \times 200 = 1000\) | 0.20 | \(1000 \times 0.20 = 200\) |
Total amount in the bag = Rs. \(200 + 150 + 200 = \) Rs. 550, which matches the given information.
The difference between the amount from 50 p coins and 20 p coins is Rs. \(200 - \) Rs. \(200 = \) Rs. 0.
Revision Table: Coin Ratio Problem
| Concept | Description | Formula/Method |
|---|---|---|
| Ratio | Expresses the relative number of items. | Let ratio parts be \(a:b:c\); actual numbers are \(ax, bx, cx\). |
| Coin Value Conversion | Converting paise to rupees. | \(1 \, \text{paise} = \text{Rs.} \, \frac{1}{100}\) |
| Total Value Calculation | Sum of values from all types of coins. | Total Value = \(\sum\) (Number of coins of type i \(\times\) Value of coin of type i) |
| Solving for Multiplier | Finding the value of \(x\) from the total value equation. | Set up equation: \(\sum (\text{Ratio} \times \text{Value} \times x) = \text{Total Amount}\), then solve for \(x\). |
| Finding Difference | Subtracting one calculated amount from another. | Difference = Amount 1 - Amount 2 |
Additional Information: Ratio and Proportion in Coin Problems
Coin problems frequently involve ratios and proportions. Understanding how to use a common multiplier (\(x\)) with the given ratio is key to solving these types of questions. Once you find the multiplier \(x\), you can determine the exact number of coins of each denomination and subsequently calculate their total value. Always remember to be consistent with units, either converting everything to paise or everything to rupees before setting up the total value equation.
These problems test your ability to translate a word problem into a mathematical equation and solve it systematically. Paying close attention to the units (paise vs. rupees) is crucial to avoid errors.
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