`22(1/2)%` of a number is 45, what per cent of that number is 90 ?

Options

A. `25%`
B. `65%`
C. `30%`
D. `45%`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the problem
We need to find out what percentage of a number (let's call it \( x \)) is equal to 90, given that \( 22 \frac{1}{2} \% \) of that number is equal to 45.

### Step 2: Convert the mixed fraction to an improper fraction
First, we convert \( 22 \frac{1}{2} \% \) to an improper fraction:
\[ 22 \frac{1}{2} = \frac{45}{2} \]
So, \( 22 \frac{1}{2} \% = \frac{45}{2} \% = \frac{45}{2} \times \frac{1}{100} = \frac{45}{200} \).

### Step 3: Set up the equation
According to the problem:
\[ \frac{45}{200} \times x = 45 \]

### Step 4: Solve for \( x \)
To find \( x \), we can rearrange the equation:
\[ x = 45 \div \left(\frac{45}{200}\right) \]
This simplifies to:
\[ x = 45 \times \frac{200}{45} \]
The \( 45 \) cancels out:
\[ x = 200 \]

### Step 5: Find what percentage 90 is of \( x \)
Now we need to find what percentage 90 is of 200:
\[ \text{Percentage} = \left(\frac{90}{200}\right) \times 100 \]

### Step 6: Calculate the percentage
Calculating this gives:
\[ \text{Percentage} = \frac{90 \times 100}{200} = \frac{9000}{200} = 45\% \]

### Conclusion
Thus, \( 90 \) is \( 45\% \) of the number \( 200 \).

### Final Answer
The required percentage is \( 45\% \).

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