A bag contains ₹310 in the form of 5 rupee, 2 rupee and 1 rupee coins in the ratio 4 ∶ 3 ∶ 5. What is the number of 5 rupee coins?
40
Understanding the Coin Ratio Problem
This problem involves a bag containing coins of different denominations: ₹5, ₹2, and ₹1. We are given the total value of the coins in the bag and the ratio of the number of coins of each denomination. Our goal is to find out the exact number of ₹5 coins.
The given information is:
- Total value in the bag: ₹310
- Denominations of coins: ₹5, ₹2, and ₹1
- Ratio of the number of 5 rupee : 2 rupee : 1 rupee coins = 4 : 3 : 5
Let's break down the problem using the given ratio.
Setting up the Number of Coins by Ratio
The ratio of the number of coins is 4 : 3 : 5 for ₹5, ₹2, and ₹1 coins, respectively. This means that for some common multiplier, let's call it \(x\), the actual number of coins are:
- Number of ₹5 coins = \(4x\)
- Number of ₹2 coins = \(3x\)
- Number of ₹1 coins = \(5x\)
Calculating the Total Value from Each Denomination
Now, we need to find the total value contributed by each type of coin. We do this by multiplying the number of coins by their respective denominations.
- Value from ₹5 coins = (Number of ₹5 coins) \(\times\) (Denomination of ₹5 coin) = \(4x \times 5 = 20x\) rupees
- Value from ₹2 coins = (Number of ₹2 coins) \(\times\) (Denomination of ₹2 coin) = \(3x \times 2 = 6x\) rupees
- Value from ₹1 coins = (Number of ₹1 coins) \(\times\) (Denomination of ₹1 coin) = \(5x \times 1 = 5x\) rupees
Forming an Equation for the Total Value
The total value in the bag is the sum of the values contributed by each type of coin. We are given that the total value is ₹310.
Total Value = Value from ₹5 coins + Value from ₹2 coins + Value from ₹1 coins
Total Value = \(20x + 6x + 5x\)
We can combine the terms on the right side:
Total Value = \((20 + 6 + 5)x = 31x\)
Since the total value is ₹310, we can set up the equation:
\(31x = 310\)
Solving for the Common Multiplier (x)
To find the value of \(x\), we need to divide the total value by the sum of the ratio parts weighted by their denominations:
\(x = \frac{310}{31}\)
\(x = 10\)
Finding the Number of 5 Rupee Coins
We determined earlier that the number of ₹5 coins is \(4x\). Now that we know \(x=10\), we can find the actual number of ₹5 coins:
Number of ₹5 coins = \(4x = 4 \times 10 = 40\)
So, there are 40 five-rupee coins in the bag.
Verification (Optional but Recommended)
Let's check if the total value with 40 ₹5 coins, \(3x = 3 \times 10 = 30\) ₹2 coins, and \(5x = 5 \times 10 = 50\) ₹1 coins is indeed ₹310.
- Value from ₹5 coins = \(40 \times 5 = 200\) rupees
- Value from ₹2 coins = \(30 \times 2 = 60\) rupees
- Value from ₹1 coins = \(50 \times 1 = 50\) rupees
Total Value = \(200 + 60 + 50 = 310\) rupees.
This matches the given total value, so our calculation is correct.
Final Answer for 5 Rupee Coins
The number of 5 rupee coins in the bag is 40.
| Denomination | Ratio Part | Number of Coins (\(x=10\)) | Value (₹) |
|---|---|---|---|
| ₹5 | 4 | \(4 \times 10 = 40\) | \(40 \times 5 = 200\) |
| ₹2 | 3 | \(3 \times 10 = 30\) | \(30 \times 2 = 60\) |
| ₹1 | 5 | \(5 \times 10 = 50\) | \(50 \times 1 = 50\) |
| Total | \(4+3+5=12\) parts | \(40+30+50=120\) coins | \(200+60+50=310\) rupees |
Revision Table: Key Concepts in Coin Ratio Problems
| Concept | Explanation | How it applies here |
|---|---|---|
| Ratio | Represents the relative number of items. If a:b:c, actual numbers are ax, bx, cx for some multiplier x. | Number of coins are 4x, 3x, 5x. |
| Total Value Calculation | Sum of (Number of items of a type × Value per item) for all types. | Sum of (Number of 5 rupee coins × ₹5) + (Number of 2 rupee coins × ₹2) + (Number of 1 rupee coins × ₹1). |
| Setting up Equation | Equating the calculated total value (in terms of x) to the given total value. | \(31x = 310\). |
| Solving for Multiplier | Finding the value of x from the equation. | \(x = 10\). |
| Finding Specific Quantity | Substituting the value of x back into the expression for the required quantity. | Number of 5 rupee coins = \(4x = 4 \times 10 = 40\). |
Additional Information: Ratio and Proportion Basics
A ratio is a way to compare two or more quantities of the same kind. For example, the ratio 4:3 means that the first quantity is 4 parts and the second is 3 parts. If the ratio of three numbers is a:b:c, it implies that the numbers are in the form \(ax\), \(bx\), and \(cx\) for some non-zero number \(x\). This \(x\) is the common multiplier that scales the ratio to the actual quantities.
In problems involving ratios and values, it's crucial to distinguish between the ratio of the number of items and the ratio of their values. Here, the ratio 4:3:5 is for the number of coins. The ratio of their values would be different (Value of ₹5 coins : Value of ₹2 coins : Value of ₹1 coins = \(20x : 6x : 5x = 20 : 6 : 5\)). Always make sure which ratio is provided in the problem statement.
Ratio problems often involve finding this common multiplier \(x\) by using some total or difference provided in the question, such as the total number of items, the total value, or the difference between two quantities.
Important Questions from Ratio and Proportion
If the ratio of three numbers A, B and C is 2 ∶ 3 ∶ 5, and the sum of the squares of these numbers is 3800, then the value of C is:
The ratio of three numbers is 3 ∶ 5 ∶ 4 and the sum of their squares is 11250. Find the sum of the numbers.
When 'x' is subtracted from each of the numbers 22, 39, 56 and 107, then the resulting numbers, in this order, are in proportion. What is the mean proportional between (x + 3) and (3x - 7)?
The salaries of Ravi and Sumit are in the ratio 4 ∶ 5. If the salary of each is increased by Rs. 6,000 the new ratio becomes 35 ∶ 40. What will be Sumit's increased salary?
u : v = 4 : 7 and v : w = 9 : 7. If u = 72, then what is the value of w?
