Two numbers are respectively 20% and 50% more than the third number. What % is the first number of the second ?

Options

A. a) 40%
B. b) 50%
C. c) 80%
D. d) 70%

Answer

To solve the problem step by step, let's define the numbers clearly and calculate the required percentage.

### Step 1: Define the third number
Let the third number be \( C \).

### Step 2: Calculate the first number
The first number \( A \) is 20% more than the third number \( C \).
This can be expressed as:
\[ A = C + 20\% \text{ of } C \]
Calculating 20% of \( C \):
\[ 20\% \text{ of } C = \frac{20}{100} \times C = \frac{1}{5}C \]
So, we can write:
\[ A = C + \frac{1}{5}C = \frac{5}{5}C + \frac{1}{5}C = \frac{6}{5}C \]

### Step 3: Calculate the second number
The second number \( B \) is 50% more than the third number \( C \).
This can be expressed as:
\[ B = C + 50\% \text{ of } C \]
Calculating 50% of \( C \):
\[ 50\% \text{ of } C = \frac{50}{100} \times C = \frac{1}{2}C \]
So, we can write:
\[ B = C + \frac{1}{2}C = \frac{2}{2}C + \frac{1}{2}C = \frac{3}{2}C \]

### Step 4: Find what percentage the first number is of the second number
We need to find what percentage \( A \) is of \( B \):
\[ \text{Percentage} = \left( \frac{A}{B} \right) \times 100 \]
Substituting the values of \( A \) and \( B \):
\[ \text{Percentage} = \left( \frac{\frac{6}{5}C}{\frac{3}{2}C} \right) \times 100 \]
The \( C \) cancels out:
\[ \text{Percentage} = \left( \frac{\frac{6}{5}}{\frac{3}{2}} \right) \times 100 \]
To divide fractions, we multiply by the reciprocal:
\[ \text{Percentage} = \left( \frac{6}{5} \times \frac{2}{3} \right) \times 100 \]
Calculating the multiplication:
\[ \text{Percentage} = \left( \frac{12}{15} \right) \times 100 \]
Simplifying \( \frac{12}{15} \):
\[ \frac{12}{15} = \frac{4}{5} \]
Now, substituting back:
\[ \text{Percentage} = \left( \frac{4}{5} \right) \times 100 = 80\% \]

### Final Answer
The first number is 80% of the second number.

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