Understanding the Salary Changes

This problem involves calculating the ratio of initial salaries of two employees, X and Y, after their salaries undergo a series of percentage changes (increases and decreases).

Calculating Employee X's Final Salary

Let the initial salary of employee X be \(S_X\).

• First, X's salary is increased by 12%. The new salary becomes: \(S_X \times (1 + \frac{12}{100}) = S_X \times (1 + 0.12) = 1.12 \times S_X\)
• Next, this new salary is decreased by 10%. The final salary of X is: \((1.12 \times S_X) \times (1 - \frac{10}{100}) = (1.12 \times S_X) \times (1 - 0.10) = (1.12 \times S_X) \times 0.90\)
• Final Salary of X = \(1.008 \times S_X\)

Calculating Employee Y's Final Salary

Let the initial salary of employee Y be \(S_Y\).

• First, Y's salary is increased by 10%. The new salary becomes: \(S_Y \times (1 + \frac{10}{100}) = S_Y \times (1 + 0.10) = 1.10 \times S_Y\)
• Next, this new salary is decreased by 12%. The final salary of Y is: \((1.10 \times S_Y) \times (1 - \frac{12}{100}) = (1.10 \times S_Y) \times (1 - 0.12) = (1.10 \times S_Y) \times 0.88\)
• Final Salary of Y = \(0.968 \times S_Y\)

Determining the Ratio of Initial Salaries

The problem states that their final salaries are equal. Therefore:

Final Salary of X = Final Salary of Y

\(1.008 \times S_X = 0.968 \times S_Y\)

To find the ratio of the initial salary of X to the initial salary of Y (\(S_X : S_Y\)), we rearrange the equation:

\(\frac{S_X}{S_Y} = \frac{0.968}{1.008}\)

Now, we simplify this fraction. We can remove the decimal points by multiplying the numerator and denominator by 1000:

\(\frac{S_X}{S_Y} = \frac{968}{1008}\)

To simplify further, we find the greatest common divisor or divide by common factors. Both numbers are divisible by 8:

• \(968 \div 8 = 121\)
• \(1008 \div 8 = 126\)

So, the simplified ratio is:

\(\frac{S_X}{S_Y} = \frac{121}{126}\)

Therefore, the ratio of the initial salary of X to the initial salary of Y was 121 : 126.