A bag contains Rs. 420 in the form of 5 rupee, 2 rupee, and 1 rupee coins. The number of coins of 5 rupee, 2 rupee, and 1 rupee is in the ratio of 2 : 3 : 5. What is the total number of coins in the bag?
200
Understanding the Coin Ratio Problem
The problem describes a bag containing coins of three different denominations: 5 rupee, 2 rupee, and 1 rupee. We are given the total value of all these coins and the ratio of the number of coins for each denomination. We need to find the total number of coins in the bag.
Let's break down the information given:
- Total value of coins = Rs. 420
- Denominations of coins: Rs. 5, Rs. 2, Re. 1
- Ratio of the number of coins (5 rupee : 2 rupee : 1 rupee) = 2 : 3 : 5
Since the number of coins is in the ratio 2 : 3 : 5, we can represent the actual number of coins of each denomination using a common multiple, let's call it \(x\).
- Number of 5 rupee coins = \(2x\)
- Number of 2 rupee coins = \(3x\)
- Number of 1 rupee coins = \(5x\)
Calculating Value from Each Coin Type
To find the total value, we need to calculate the value contributed by the coins of each denomination and sum them up. The value from a set of coins is calculated by multiplying the number of coins by the value of each coin.
- Value from 5 rupee coins = (Number of 5 rupee coins) \(\times\) (Value of each 5 rupee coin) = \(2x \times 5 = 10x\) rupees
- Value from 2 rupee coins = (Number of 2 rupee coins) \(\times\) (Value of each 2 rupee coin) = \(3x \times 2 = 6x\) rupees
- Value from 1 rupee coins = (Number of 1 rupee coins) \(\times\) (Value of each 1 rupee coin) = \(5x \times 1 = 5x\) rupees
Solving for the Unknown Factor in the Coin Ratio
The total value of the coins in the bag is the sum of the values from each denomination. We are given that the total value is Rs. 420. So, we can set up an equation:
Total value = Value from 5 rupee coins + Value from 2 rupee coins + Value from 1 rupee coins
\(420 = 10x + 6x + 5x\)
Combine the terms on the right side of the equation:
\(420 = (10 + 6 + 5)x\)
\(420 = 21x\)
Now, we can solve for \(x\) by dividing the total value by 21:
\(x = \frac{420}{21}\)
\(x = 20\)
The value of \(x\) is 20. This factor helps us determine the actual number of coins of each type.
Determining the Total Number of Coins
Now that we have the value of \(x\), we can find the actual number of coins of each denomination:
- Number of 5 rupee coins = \(2x = 2 \times 20 = 40\) coins
- Number of 2 rupee coins = \(3x = 3 \times 20 = 60\) coins
- Number of 1 rupee coins = \(5x = 5 \times 20 = 100\) coins
To find the total number of coins in the bag, we sum the number of coins of all denominations:
Total number of coins = Number of 5 rupee coins + Number of 2 rupee coins + Number of 1 rupee coins
Total number of coins = \(40 + 60 + 100\)
Total number of coins = \(200\)
So, the total number of coins in the bag is 200.
| Coin Denomination | Ratio | Number of Coins (\(x=20\)) | Value per Coin (Rs.) | Total Value (Rs.) |
|---|---|---|---|---|
| 5 Rupee | 2 | \(2 \times 20 = 40\) | 5 | \(40 \times 5 = 200\) |
| 2 Rupee | 3 | \(3 \times 20 = 60\) | 2 | \(60 \times 2 = 120\) |
| 1 Rupee | 5 | \(5 \times 20 = 100\) | 1 | \(100 \times 1 = 100\) |
| Total | \(40 + 60 + 100 = 200\) | \(200 + 120 + 100 = 420\) | ||
The calculated total value (Rs. 420) matches the given total value, which confirms our calculations for the number of coins are correct.
Revision Table: Coin Ratio Problem Solving
| Step | Description | Calculation/Concept |
|---|---|---|
| 1 | Represent number of coins using ratio and a variable (\(x\)). | 5 rupee: \(2x\), 2 rupee: \(3x\), 1 rupee: \(5x\) |
| 2 | Calculate value from each coin type. | 5 rupee: \(2x \times 5 = 10x\) 2 rupee: \(3x \times 2 = 6x\) 1 rupee: \(5x \times 1 = 5x\) |
| 3 | Set up equation using total value. | \(10x + 6x + 5x = 420 \implies 21x = 420\) |
| 4 | Solve for the variable \(x\). | \(x = \frac{420}{21} = 20\) |
| 5 | Calculate the actual number of coins for each type. | 5 rupee: \(2 \times 20 = 40\) 2 rupee: \(3 \times 20 = 60\) 1 rupee: \(5 \times 20 = 100\) |
| 6 | Calculate the total number of coins. | \(40 + 60 + 100 = 200\) |
Additional Information on Ratio and Proportion Problems
Ratio problems often involve distributing a total quantity or value according to given ratios. A common approach is to represent the parts of the ratio using a variable, say \(x\), so the quantities are \(ax, bx, cx, \dots\) if the ratio is \(a:b:c:\dots\). The sum or difference of these quantities is then related to a given total or difference, forming an equation to solve for \(x\). Once \(x\) is found, the actual quantities can be calculated.
In this coin problem, the ratio applies to the number of coins, while the total given is a value. Therefore, we first needed to convert the number of coins in terms of \(x\) into their corresponding value in terms of \(x\), and then equate the total value expression to the given total value.
Understanding the difference between a ratio of quantities (like number of coins) and the ratio of values is crucial in solving such problems. Always ensure that the equation you set up relates quantities of the same type (either number of coins or total value).
Important Questions from Simple Ratios
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct ?
A certain number is added to each of a pair of numbers which are in the ratio 4 ∶ 5. The sum of the resulting numbers is 39 and their ratio (taken in the same order as mentioned above) is 6 ∶ 7. What is the number added?
If (p + q) ∶ (q + r) ∶ (r + p) = 5 ∶ 6 ∶ 7 and p + q + r = 18, find the value of p ∶ q ∶ r
If P/Q = 8/9 and Q - P = 12, then what is the value of P?
