In a basic percentage problem, three numbers x, y and z are such that x is 30% of z and y is 40% of z. If x is equal to p% of y, what is the value of p?

Introduction / Context:
This question tests the concept of percentages as ratios between related quantities. It focuses on how one number can be expressed as a percentage of another when both are defined in terms of a third base number. Such questions are very common in quantitative aptitude tests and competitive exams, because they check conceptual clarity rather than heavy calculation.

Given Data / Assumptions:
- x is 30 percent of z.
- y is 40 percent of z.
- We need to find a value p such that x is p percent of y.
- All numbers are real and positive in the usual aptitude context.

Concept / Approach:
A percentage is simply a fraction multiplied by 100. If one quantity is a certain percent of another, we can write it as (percentage / 100) times the reference quantity. Here, x and y are both written as percentages of z, so we first express them in terms of z. Then we compare x and y to find what percent x is of y by using the ratio x / y and converting it into a percentage.

Step-by-Step Solution:
Let z be any convenient base value (for example, z itself as a symbol). Given: x is 30% of z, so x = (30 / 100) * z = 0.30z. Similarly, y is 40% of z, so y = (40 / 100) * z = 0.40z. We are told that x is p% of y. This means x = (p / 100) * y. Substitute x and y in terms of z: 0.30z = (p / 100) * 0.40z. Cancel z from both sides: 0.30 = (p / 100) * 0.40. Rearrange for p: p / 100 = 0.30 / 0.40 = 3 / 4. So p = (3 / 4) * 100 = 75.

Verification / Alternative check:
Take an easy numerical value for z, such as z = 100. Then x = 30% of 100 = 30, and y = 40% of 100 = 40. Now find what percent 30 is of 40. The fraction is 30 / 40 = 3 / 4. Converting to a percentage, (3 / 4) * 100 = 75%. This confirms our algebraic result and shows that p must be 75.

Why Other Options Are Wrong:
45: This would imply x is less than half of y, but 30 compared with 40 is more than half, so 45% is too small.
55: 55% of 40 is 22, which is less than 30, so this does not match x.
65: 65% of 40 is 26, still below 30, so it is also incorrect.
60: 60% of 40 is 24, which again does not equal x = 30.

Common Pitfalls:
A common mistake is to subtract percentages directly, for example thinking 40% minus 30% is 10%, and somehow linking that to p. Another error is to confuse the base quantity, using z instead of y when computing p. Always remember that when we say “x is p percent of y”, the base is y, not z. It is also important not to mix fraction and percent without converting correctly.

Final Answer:
Therefore, the required percentage p such that x is p percent of y is 75 percent.