The salary of Gadadhar is 3/2 times the salary of Haamid, and the salary of Sarvesh is 4/3 times the salary of Haamid. What is the ratio of Gadadhar's salary to Sarvesh's salary?

Introduction / Context:
This is a direct ratio comparison problem involving salaries of three people. We are told how the salaries of Gadadhar and Sarvesh relate to the salary of Haamid. The goal is to express the ratio of Gadadhar's salary to Sarvesh's salary. Such questions test the ability to handle fractional multipliers and convert them into a simple ratio between two quantities.

Given Data / Assumptions:

Let the salary of Haamid be some base amount H.
Salary of Gadadhar = 3/2 times H.
Salary of Sarvesh = 4/3 times H.
We need the ratio of Gadadhar's salary to Sarvesh's salary.
All salaries are positive real numbers.

Concept / Approach:
When two quantities are given as fractional multiples of a third quantity, we can write them in terms of that base variable and then form their ratio directly. Here we write Gadadhar's salary and Sarvesh's salary both in terms of H and simplify the resulting fraction. The base salary H cancels out, leaving a ratio involving only the given fractions. This method is simple and avoids any need to assume actual rupee values.

Step-by-Step Solution:
Let Haamid's salary be H. Then Gadadhar's salary = (3/2) * H. Sarvesh's salary = (4/3) * H. Required ratio = Gadadhar's salary : Sarvesh's salary. So ratio = (3/2) * H : (4/3) * H. Cancel H from both terms, giving (3/2) : (4/3). Write the ratio as a single fraction: [(3/2) / (4/3)] = (3/2) * (3/4) = 9/8. Therefore the ratio of Gadadhar's salary to Sarvesh's salary is 9 : 8.

Verification / Alternative check:
We can pick a convenient value for H to check. For example, let H = 6. Then Gadadhar's salary = (3/2) * 6 = 9 and Sarvesh's salary = (4/3) * 6 = 8. Their ratio is 9:8, which matches the derived ratio. This confirms that our algebraic simplification is correct and independent of the particular value chosen for H.

Why Other Options Are Wrong:
Option 1:2 would imply that Gadadhar earns half as much as Sarvesh, which contradicts the given multipliers 3/2 and 4/3.
Option 2:1 would mean Gadadhar earns twice as much as Sarvesh, again inconsistent with the fractional relationship provided.
Option 8:9 reverses the correct ratio and would correspond to Sarvesh earning more than Gadadhar, which does not follow from the given data.

Common Pitfalls:
Some students add the fractions 3/2 and 4/3 instead of forming the required ratio between them.
Others forget to cancel the common factor H and mistakenly think the actual value of H matters when it cancels out completely.
A few learners invert the ratio and compute (4/3) : (3/2) instead of the required (3/2) : (4/3), thus getting 8:9 instead of 9:8.

Final Answer:
The ratio of Gadadhar's salary to Sarvesh's salary is 9 : 8.