In a wallet, there are notes of the denominations Rs. 10 and Rs. 50. The total number of notes is 12. The number of Rs. 10 and Rs. 50 notes are in the ratio of 1 : 2. Total money in the wallet is:
Rs. 440
Finding Total Money in Wallet with Denominations
This problem asks us to calculate the total amount of money in a wallet containing notes of two different denominations: Rs. 10 and Rs. 50. We are given the total number of notes and the ratio of the number of notes for each denomination.
Understanding the Problem Details
- Denominations: Rs. 10 and Rs. 50.
- Total number of notes: 12.
- Ratio of Rs. 10 notes to Rs. 50 notes: 1 : 2.
Our goal is to find the total value of all notes in the wallet.
Step-by-Step Calculation
Step 1: Determine the Number of Each Type of Note
The ratio of the number of Rs. 10 notes to Rs. 50 notes is 1 : 2. This means for every 1 Rs. 10 note, there are 2 Rs. 50 notes.
Let the common ratio factor be $k$.
- Number of Rs. 10 notes $= 1 \times k = k$
- Number of Rs. 50 notes $= 2 \times k = 2k$
The total number of notes is 12. So, the sum of the number of Rs. 10 notes and Rs. 50 notes must equal 12.
\(k + 2k = 12\)
\(3k = 12\)
To find the value of $k$, divide both sides by 3:
\(k = \frac{12}{3}\)
\(k = 4\)
Now we can find the actual number of notes for each denomination:
- Number of Rs. 10 notes $= k = 4$
- Number of Rs. 50 notes $= 2k = 2 \times 4 = 8$
Let's check if the total number of notes is correct: \(4 + 8 = 12\). Yes, it matches the given total.
Step 2: Calculate the Total Value of Each Type of Note
Now that we know the number of notes for each denomination, we can calculate the total value contributed by each type of note.
- Value from Rs. 10 notes = (Number of Rs. 10 notes) $\times$ (Value of each Rs. 10 note)
- Value from Rs. 10 notes = \(4 \times 10 = 40\) Rs.
- Value from Rs. 50 notes = (Number of Rs. 50 notes) $\times$ (Value of each Rs. 50 note)
- Value from Rs. 50 notes = \(8 \times 50 = 400\) Rs.
Step 3: Calculate the Total Money in the Wallet
The total money in the wallet is the sum of the values from the Rs. 10 notes and the Rs. 50 notes.
Total money = Value from Rs. 10 notes + Value from Rs. 50 notes
Total money = \(40 + 400\)
Total money = \(440\) Rs.
The total money in the wallet is Rs. 440.
Summary of Results
| Denomination | Ratio Part | Calculated Number of Notes | Value per Note (Rs.) | Total Value (Rs.) |
|---|---|---|---|---|
| Rs. 10 | 1 | 4 | 10 | \(4 \times 10 = 40\) |
| Rs. 50 | 2 | 8 | 50 | \(8 \times 50 = 400\) |
| Total Notes: 12 | Total Money: \(40 + 400 = 440\) | |||
Revision Table: Understanding Ratios and Money Problems
| Concept | Description | How it was used here |
|---|---|---|
| Ratio | A comparison of two quantities. If the ratio is $a:b$, the quantities can be represented as $ak$ and $bk$ for some factor $k$. | The ratio 1:2 for notes meant the number of notes were $1k$ and $2k$. |
| Total Quantity | The sum of individual quantities. | The total number of notes (12) was the sum of the number of Rs. 10 notes and Rs. 50 notes ($k + 2k$). |
| Calculating Total Value | Multiply the number of items by the value per item and sum up for all types of items. | Calculated total value by (Number of Rs. 10 notes $\times$ 10) + (Number of Rs. 50 notes $\times$ 50). |
Additional Information: Solving Similar Money Problems
Problems involving different denominations of currency and ratios are common in competitive exams. Here are some tips for solving them:
- Identify Denominations and Quantities: Clearly list the different values of currency notes or coins involved and any given totals (like total number of items or total value).
- Use Variables for Unknowns: If counts or values are unknown, assign variables (like $x$, $y$, or $k$).
- Set up Equations: Translate the problem statement into mathematical equations. Ratios often lead to expressions like $ax$ and $bx$. Total counts or values provide equations like $ax + bx = \text{Total}$.
- Solve the Equations: Use algebraic methods to find the value of the variables.
- Calculate the Final Answer: Once the number of each item is known, calculate the required total value or count.
- Check Your Work: Substitute the calculated numbers back into the original problem statement to ensure they satisfy all conditions (total number of notes, total value, ratio, etc.).
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