90% of x=40% of y and y=a% of x. What is the value of a?

Options

A. 225
B. 125
C. 2.25
D. 1.25

Answer

To solve the problem step by step, we start with the equations given in the question.

### Step 1: Write down the equations
We have two equations based on the problem statement:
1. \( 90\% \text{ of } x = 40\% \text{ of } y \)
2. \( y = a\% \text{ of } x \)

### Step 2: Convert percentages to fractions
Convert the percentages in the first equation to fractions:
- \( 90\% = \frac{90}{100} = 0.9 \)
- \( 40\% = \frac{40}{100} = 0.4 \)

So, the first equation becomes:
\[ 0.9x = 0.4y \]

### Step 3: Rearranging the first equation
Rearranging the equation to express \( y \) in terms of \( x \):
\[ y = \frac{0.9x}{0.4} \]
\[ y = 2.25x \]

### Step 4: Substitute \( y \) in the second equation
Now, substitute \( y = 2.25x \) into the second equation \( y = a\% \text{ of } x \):
\[ 2.25x = a\% \text{ of } x \]
This can be expressed as:
\[ 2.25x = \frac{a}{100} \cdot x \]

### Step 5: Cancel \( x \) from both sides
Assuming \( x \neq 0 \), we can divide both sides by \( x \):
\[ 2.25 = \frac{a}{100} \]

### Step 6: Solve for \( a \)
To find \( a \), multiply both sides by 100:
\[ a = 2.25 \times 100 \]
\[ a = 225 \]

### Final Answer
Thus, the value of \( a \) is \( 225 \).

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