A person carries Rs. 165/ - in the form of currency notes of denominations Rs. 5, Rs. 10 & Rs. 20 in the ratio of 3 : 2 : 1. What is the value of currency notes of Rs. 20 denomination?
Rs. 60
Solving the Currency Note Ratio Problem
This problem involves understanding ratios and how they relate to the value of different denominations of currency notes. We are given the total amount of money and the ratio of the *number* of notes for each denomination. Our goal is to find the total value of the notes of a specific denomination (Rs. 20).
Understanding the Ratio of Currency Notes
The ratio of the number of Rs. 5, Rs. 10, and Rs. 20 notes is given as 3 : 2 : 1. This means that for some number 'x', the quantity of each type of note can be represented as:
- Number of Rs. 5 notes = $3x$
- Number of Rs. 10 notes = $2x$
- Number of Rs. 20 notes = $1x$ (or simply $x$)
Calculating the Total Value from the Ratio
To find the total value of money, we multiply the number of notes of each denomination by their respective value and sum them up. The total value is given as Rs. 165.
- Value from Rs. 5 notes = (Number of Rs. 5 notes) $\times$ 5 = $(3x) \times 5 = 15x$
- Value from Rs. 10 notes = (Number of Rs. 10 notes) $\times$ 10 = $(2x) \times 10 = 20x$
- Value from Rs. 20 notes = (Number of Rs. 20 notes) $\times$ 20 = $(x) \times 20 = 20x$
The total value is the sum of these values:
Total Value = Value from Rs. 5 notes + Value from Rs. 10 notes + Value from Rs. 20 notes
$165 = 15x + 20x + 20x$
Solving for the Unknown 'x'
Now, we need to solve the equation for $x$:
$165 = 15x + 20x + 20x$
Combine the terms on the right side:
$165 = (15 + 20 + 20)x$
$165 = 55x$
To find $x$, divide both sides by 55:
$x = \frac{165}{55}$
$x = 3$
So, the value of 'x' is 3.
Determining the Number of Each Note and Value
Using the value of $x=3$, we can find the actual number of notes of each denomination:
- Number of Rs. 5 notes = $3x = 3 \times 3 = 9$ notes
- Number of Rs. 10 notes = $2x = 2 \times 3 = 6$ notes
- Number of Rs. 20 notes = $x = 1 \times 3 = 3$ notes
Let's verify the total value:
- Value of Rs. 5 notes = $9 \times 5 = 45$
- Value of Rs. 10 notes = $6 \times 10 = 60$
- Value of Rs. 20 notes = $3 \times 20 = 60$
Total Value = $45 + 60 + 60 = 165$. This matches the given total amount.
Finding the Value of Rs. 20 Denomination Notes
The question asks for the value of currency notes of Rs. 20 denomination. From our calculation above:
Value of Rs. 20 notes = (Number of Rs. 20 notes) $\times$ 20
Value of Rs. 20 notes = $3 \times 20 = 60$
The value of Rs. 20 denomination notes is Rs. 60.
Summary of Currency Notes and Value
| Denomination | Ratio Part | Number of Notes ($x=3$) | Value per Note (Rs.) | Total Value (Rs.) |
|---|---|---|---|---|
| Rs. 5 | 3 | $3 \times 3 = 9$ | 5 | $9 \times 5 = 45$ |
| Rs. 10 | 2 | $2 \times 3 = 6$ | 10 | $6 \times 10 = 60$ |
| Rs. 20 | 1 | $1 \times 3 = 3$ | 20 | $3 \times 20 = 60$ |
| Total Value | $45 + 60 + 60 = 165$ | |||
Conclusion on Currency Denomination Value
Based on the ratio and the total amount, the value contributed by the Rs. 20 notes is Rs. 60.
Revision Table: Currency Ratio Problem
| Concept | Explanation |
|---|---|
| Ratio | A comparison of two or more quantities. Here, it's the ratio of the *number* of notes (3:2:1). |
| Representing Ratio | Use a variable ($x$) to represent parts of the ratio. Number of notes = $3x, 2x, x$. |
| Calculating Value | Value = (Number of items) $\times$ (Value per item). For currency, Value = (Number of notes) $\times$ (Denomination). |
| Setting up Equation | Sum of values from each denomination equals the total given amount. $15x + 20x + 20x = 165$. |
| Solving for Variable | Simplify the equation and isolate the variable ($x$). $55x = 165 \implies x=3$. |
| Finding Specific Value | Use the variable's value to find the number of specific notes and then calculate their total value. Number of Rs. 20 notes = $x=3$; Value = $3 \times 20 = 60$. |
Additional Information: Ratio and Proportion in Problems
Ratio and proportion are fundamental concepts used to solve various types of problems involving comparisons and distributions. In this currency note problem, the ratio of the *number* of notes is given, but we work with the *value* contributed by each type of note to find the unknown multiplier 'x'.
- Ratio: Expresses how many times one number contains another. A ratio a:b means a/b.
- Proportion: An equality between two ratios. If a:b = c:d, then ad = bc.
- Applying Ratios in Problems: When a quantity is divided in a ratio a:b:c, the parts can be written as $ax, bx, cx$. The sum of the parts is equal to the total quantity: $ax + bx + cx =$ Total.
- Ratio vs. Value: It's crucial to distinguish between the ratio of quantities (like the number of notes) and the ratio or sum of their values. In this problem, the ratio is of the count of notes, not their worth.
- Unitary Method: Once 'x' is found, it acts like a unit. $3x$ notes mean 3 units of Rs. 5 notes, $2x$ notes mean 2 units of Rs. 10 notes, and $x$ notes mean 1 unit of Rs. 20 notes. Each 'unit' of notes here corresponds to 3 actual notes ($x=3$).
Understanding how to convert ratios of items into ratios of their values (or total value) is key to solving such problems effectively.
Important Questions from Compound Ratios
The sum of three numbers is 280. If the ratio between the first and second numbers is 2 : 3 and the ratio between second and third numbers is 4 : 5, find the second number.
When x is subtracted from each of the numbers 54, 49, 22 and 21, the numbers so obtained are in proportion. The ratio of (8x - 25) to (7x - 26) is:
The train fare, bus fare and air fare between 2 places are in the ratio 5 : 8 : 12, the number of passenger travelled by them is in the ratio 3 : 4 : 5 and the total fare collected on a particular day for these modes of transportation for a single trip is Rs. 1,07,000. Find the fare collected from the air passengers.
If a: b = 5: 3, then (a³-b³): (a³+b³) = ?
What is the compound ratio of 2 : 3, 4 : 7 and 5 : 6?
