One year ago, the ratio between A's and B's salary was `4 : 5`. The ratio of their individual salaries of last year and present year are `3 : 5` and `2 : 3` respectively. If their total salaries for the present year is Rs 68., the present salary of A is

Options

A. Rs 4080
B. Rs 3200
C. Rs `4533. 40`
D. Rs 2720

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the Salaries One Year Ago
Let A's salary one year ago be \(4k\) and B's salary one year ago be \(5k\), based on the ratio \(4:5\).

### Step 2: Define the Salaries Last Year and Present Year
According to the problem:
- The ratio of A's last year salary to present salary is \(3:5\).
- The ratio of B's last year salary to present salary is \(2:3\).

Let:
- A's last year salary = \(3x\) and A's present salary = \(5x\).
- B's last year salary = \(2y\) and B's present salary = \(3y\).

### Step 3: Relate the Salaries
From the information given, we can set up the following equations based on the salaries one year ago:
1. From A's salary:
\[ 3x = 4k \quad \text{(A's last year salary)} \]
2. From B's salary:
\[ 2y = 5k \quad \text{(B's last year salary)} \]

### Step 4: Express \(k\) in Terms of \(x\) and \(y\)
From the first equation:
\[ k = \frac{3x}{4} \]
From the second equation:
\[ k = \frac{2y}{5} \]

### Step 5: Set the Equations Equal
Since both expressions equal \(k\), we can set them equal to each other:
\[ \frac{3x}{4} = \frac{2y}{5} \]

### Step 6: Cross Multiply to Solve for \(y\) in Terms of \(x\)
Cross multiplying gives:
\[ 3x \cdot 5 = 2y \cdot 4 \]
\[ 15x = 8y \quad \text{(Equation 1)} \]

### Step 7: Use the Total Salary Information
The total present salary is given as Rs 68,000:
\[ 5x + 3y = 68000 \quad \text{(Equation 2)} \]

### Step 8: Substitute \(y\) from Equation 1 into Equation 2
From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = \frac{15x}{8} \]
Substituting this into Equation 2:
\[ 5x + 3\left(\frac{15x}{8}\right) = 68000 \]
\[ 5x + \frac{45x}{8} = 68000 \]

### Step 9: Combine Like Terms
To combine the terms, convert \(5x\) to have a common denominator:
\[ \frac{40x}{8} + \frac{45x}{8} = 68000 \]
\[ \frac{85x}{8} = 68000 \]

### Step 10: Solve for \(x\)
Multiply both sides by 8:
\[ 85x = 544000 \]
Now divide by 85:
\[ x = \frac{544000}{85} = 6400 \]

### Step 11: Find A's Present Salary
Now that we have \(x\), we can find A's present salary:
\[ A's \, present \, salary = 5x = 5 \times 6400 = 32000 \]

### Conclusion
Thus, the present salary of A is Rs 32,000.

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