There are three numbers. The first number is `33^(1/3)%` more than the second number and `50%` more than the third number. What percentage of the third number is the second number ?

Options

A. `107.5%`
B. `112.5%
C. `117.5%`
D. `112.5%`

Explanation

To solve the problem step by step, we will denote the three numbers as follows:

- Let the second number be \( S \).
- The first number is \( 33^{1/3}\% \) more than the second number.
- The first number is also \( 50\% \) more than the third number.

### Step 1: Express the first number in terms of the second number
The first number \( F \) can be expressed as:
\[ F = S + \left( \frac{33^{1/3}}{100} \times S \right) \]
Calculating \( 33^{1/3} \):
\[ 33^{1/3} = \frac{100}{3} \text{ (approximately 33.33)} \]
Thus,
\[ F = S + \left( \frac{33.33}{100} \times S \right) = S + 0.3333S = \frac{4}{3}S \]

### Step 2: Express the third number in terms of the first number
The first number \( F \) is also \( 50\% \) more than the third number \( T \):
\[ F = T + \left( \frac{50}{100} \times T \right) = T + 0.5T = 1.5T \]
From this, we can express \( T \) in terms of \( F \):
\[ T = \frac{F}{1.5} = \frac{2}{3}F \]

### Step 3: Substitute the expression for \( F \) into the equation for \( T \)
Now we substitute \( F \) from Step 1 into the equation for \( T \):
\[ T = \frac{2}{3} \left( \frac{4}{3}S \right) = \frac{8}{9}S \]

### Step 4: Find the percentage of the second number in terms of the third number
We need to find what percentage the second number \( S \) is of the third number \( T \):
\[ \text{Percentage} = \left( \frac{S}{T} \right) \times 100 \]
Substituting \( T \) from Step 3:
\[ \text{Percentage} = \left( \frac{S}{\frac{8}{9}S} \right) \times 100 = \left( \frac{9}{8} \right) \times 100 = 112.5\% \]

### Final Answer
The second number is \( 112.5\% \) of the third number.