Find two numbers such that the mean proportional between them is 12 and the third proportional to them is 96.

Answer

To find two numbers such that the mean proportional between them is 12 and the third proportional to them is 96, we can follow these steps:

### Step 1: Define the two numbers
Let the two numbers be \( A \) and \( B \).

### Step 2: Use the mean proportional condition
According to the problem, the mean proportional between \( A \) and \( B \) is 12. This means:
\[ \frac{A}{12} = \frac{12}{B} \]
Cross-multiplying gives us:
\[ AB = 12 \times 12 = 144 \]
This is our **Equation 1**.

### Step 3: Use the third proportional condition
The problem also states that the third proportional to \( A \) and \( B \) is 96. This means:
\[ \frac{A}{B} = \frac{B}{96} \]
Cross-multiplying gives us:
\[ A \cdot 96 = B^2 \]
This is our **Equation 2**.

### Step 4: Substitute \( A \) from Equation 1 into Equation 2
From Equation 1, we have:
\[ A = \frac{144}{B} \]
Substituting this value of \( A \) into Equation 2:
\[ \frac{144}{B} \cdot 96 = B^2 \]
This simplifies to:
\[ \frac{13824}{B} = B^2 \]
Multiplying both sides by \( B \) gives:
\[ 13824 = B^3 \]

### Step 5: Solve for \( B \)
To find \( B \), we take the cube root of both sides:
\[ B = \sqrt[3]{13824} \]
Calculating the cube root:
\[ B = 24 \]

### Step 6: Find \( A \) using \( B \)
Now substituting \( B = 24 \) back into Equation 1 to find \( A \):
\[ A \cdot 24 = 144 \]
Thus:
\[ A = \frac{144}{24} = 6 \]

### Final Answer
The two numbers are \( A = 6 \) and \( B = 24 \).

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