Solving the Officer and Clerk Salary Ratio Problem

This problem involves understanding and applying the concept of ratios to find the initial salaries of an officer and a clerk based on their initial ratio, salary increases, and the resulting new ratio.

Understanding the Initial Salary Ratio

The initial monthly salaries of the officer and the clerk are given in the ratio 11 : 4. This means that for some common factor, let's call it \(x\), the officer's initial salary can be represented as \(11x\) and the clerk's initial salary as \(4x\).

• Initial Officer's Salary = \(11x\)
• Initial Clerk's Salary = \(4x\)

Accounting for Salary Increases

Next, the salaries increase. The officer's salary increases by ₹7,000, and the clerk's salary increases by ₹3,000.

• Officer's new salary = Initial Officer's Salary + ₹7,000 = \(11x + 7000\)
• Clerk's new salary = Initial Clerk's Salary + ₹3,000 = \(4x + 3000\)

Setting Up the Equation Based on the New Ratio

After the increases, the new ratio of the officer's salary to the clerk's salary becomes 19 : 7. We can write this as an equation:

$$ \frac{\text{New Officer's Salary}}{\text{New Clerk's Salary}} = \frac{19}{7} $$

Substituting the expressions for the new salaries:

$$ \frac{11x + 7000}{4x + 3000} = \frac{19}{7} $$

Solving the Equation to Find the Value of x

To find the value of \(x\), we can cross-multiply the equation:

$$ 7 \times (11x + 7000) = 19 \times (4x + 3000) $$

Now, distribute the numbers on both sides:

$$ 77x + 49000 = 76x + 57000 $$

Collect the terms with \(x\) on one side and the constant terms on the other side:

$$ 77x - 76x = 57000 - 49000 $$

Simplify both sides:

$$ x = 8000 $$

The common factor \(x\) is 8000.

Calculating the Initial Salary of the Officer

The question asks for the initial salary of the officer. We defined the initial officer's salary as \(11x\).

Initial Officer's Salary = \(11 \times x\)

Substitute the value of \(x = 8000\):

Initial Officer's Salary = \(11 \times 8000 = 88000\)

So, the initial salary of the officer was ₹88,000.

Verification of the Solution

Let's quickly check if our value of \(x\) is correct by calculating the new salaries and checking the new ratio:

• Initial Officer's Salary = ₹88,000
• Initial Clerk's Salary = \(4x = 4 \times 8000 = ₹32,000\)
• Officer's new salary = ₹88,000 + ₹7,000 = ₹95,000
• Clerk's new salary = ₹32,000 + ₹3,000 = ₹35,000
• New Ratio = ₹95,000 : ₹35,000

Dividing both parts of the new ratio by 5000:

$$ \frac{95000}{5000} : \frac{35000}{5000} = 19 : 7 $$

This matches the given new ratio, confirming our calculation is correct.

Conclusion

The initial salary of the officer was ₹88,000.

Revision Table: Salary Ratio Problem Steps

Additional Information: Ratios and Proportions

A ratio is a comparison of two quantities by division. For example, a ratio of 11:4 means the first quantity is \(\frac{11}{4}\) times the second quantity.

When ratios are set equal to each other, it forms a proportion. In this salary problem, we used the initial ratio to set up initial salaries with a variable and then used the new ratio to set up a proportion (an equation) involving the new salaries.

Solving problems involving ratios and changes often requires setting up algebraic equations based on the given information. The key is to represent the unknown quantities using variables based on the initial ratio and then translate the changes and the new ratio into an equation that can be solved for the variable.