The mean proportion of two numbers is 24 and their third proportion is 72. Find the sum of the two numbers.

Options

A. 11
B. 24
C. 32
D. 80

Explanation

To solve the problem, we need to find two numbers \( A \) and \( B \) given their mean proportion and third proportion. Let's break it down step-by-step.

### Step 1: Understand the definitions
- The **mean proportion** of two numbers \( A \) and \( B \) is given as \( 24 \). This means:
\[ \frac{A}{B} = \frac{B}{C} \]
where \( C \) is the third proportion.

- The **third proportion** of \( A \) and \( B \) is given as \( 72 \). This means:
\[ \frac{A}{B} = \frac{B}{72} \]

### Step 2: Set up the equations
From the mean proportion, we can write:
\[ B^2 = A \cdot C \]
Given that the mean proportion is \( 24 \):
\[ B^2 = 24 \times 24 = 576 \]

From the third proportion, we know:
\[ C = 72 \]
Thus:
\[ A \cdot 72 = 576 \]

### Step 3: Solve for \( A \)
Now, we can solve for \( A \):
\[ A = \frac{576}{72} \]
Calculating this gives:
\[ A = 8 \]

### Step 4: Find \( B \)
Now that we have \( A \), we can find \( B \) using the mean proportion:
\[ B^2 = A \cdot C = 8 \cdot 72 \]
Calculating this gives:
\[ B^2 = 576 \]
Taking the square root:
\[ B = 24 \]

### Step 5: Calculate the sum of \( A \) and \( B \)
Now we can find the sum of the two numbers:
\[ A + B = 8 + 72 = 80 \]

### Final Answer
Thus, the sum of the two numbers is \( 80 \).