It is given that 90% of X is equal to 40% of Y, and also that Y is equal to a% of X. What is the value of a?

Introduction / Context:
This question is about relating two variables X and Y using percentage statements and then finding the exact percentage relationship between them. It checks whether you can translate percentage equations into algebraic form and solve for an unknown percentage factor a.

Given Data / Assumptions:

• 90 percent of X is equal to 40 percent of Y.
• Y is equal to a percent of X.
• We need to find the value of a.
• Both X and Y are assumed to be positive real numbers.

Concept / Approach:
Expressions like 90 percent of X and 40 percent of Y are written as: 90 percent of X = 90/100 * X = 0.90 * X. 40 percent of Y = 40/100 * Y = 0.40 * Y. The equation 90 percent of X = 40 percent of Y becomes: 0.90 * X = 0.40 * Y. From this, we obtain Y in terms of X, and then interpret Y as a percent of X to read off a.

Step-by-Step Solution:
Step 1: Write the given relationship as an equation: 0.90 * X = 0.40 * Y. Step 2: Solve for Y in terms of X: Y = (0.90 / 0.40) * X. Step 3: Simplify 0.90 / 0.40 = 9/4 = 2.25. Step 4: Hence, Y = 2.25 * X. Step 5: The statement "Y = a percent of X" means Y = (a / 100) * X. Step 6: Compare the two expressions for Y: (a / 100) * X = 2.25 * X. Step 7: Cancel X on both sides to get a / 100 = 2.25. Step 8: Multiply both sides by 100 to find a: a = 2.25 * 100 = 225.

Verification / Alternative check:
Choose a convenient number for X, such as X = 100. Then 90 percent of X = 90. So 40 percent of Y must be 90, giving Y = 90 / 0.40 = 225. Now Y as a percent of X = 225 / 100 * 100 percent = 225 percent. Thus a = 225, confirming the algebraic solution.

Why Other Options Are Wrong:
125: This would correspond to Y being only 1.25 times X, which does not satisfy the given relation 0.90X = 0.40Y. 2.25: This is the multiplier, not the percentage. The question specifically asks for a, which is a percent value, not a plain factor. 1.25: Also a multiplier, and too small in comparison with the correct factor 2.25. 2250: This is ten times larger than the correct percentage and does not match any logical interpretation of the relationship between X and Y.

Common Pitfalls:
One common error is to forget that a percent is obtained by multiplying the decimal factor by 100, so students may choose 2.25 instead of 225. Another mistake is to mix up which variable is expressed as a percent of the other. Always translate the wording carefully and keep track of which quantity is being described as a percent of which base.

Final Answer:
The value of a is 225, so Y is 225 percent of X.