Find two numbers such that the mean proportional between them is `14` and third proportional to them is `112`.

Answer

To find two numbers such that the mean proportional between them is 14 and the third proportional to them is 112, we can follow these steps:

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

### Step 2: Use the Mean Proportional Condition
The mean proportional between \( A \) and \( B \) is given as 14. According to the definition of mean proportion, we have:
\[ \frac{A}{14} = \frac{14}{B} \]
Cross-multiplying gives us:
\[ AB = 14 \times 14 \]
Calculating the right-hand side:
\[ AB = 196 \]
This is our first equation:
\[ (1) \quad AB = 196 \]

### Step 3: Use the Third Proportional Condition
The third proportional to \( A \) and \( B \) is given as 112. According to the definition of third proportional, we have:
\[ \frac{A}{B} = \frac{B}{112} \]
Cross-multiplying gives us:
\[ B^2 = 112A \]
This is our second equation:
\[ (2) \quad B^2 = 112A \]

### Step 4: Substitute for A
From equation (1), we can express \( A \) in terms of \( B \):
\[ A = \frac{196}{B} \]
Now, substitute this expression for \( A \) into equation (2):
\[ B^2 = 112 \left(\frac{196}{B}\right) \]
This simplifies to:
\[ B^2 = \frac{112 \times 196}{B} \]
Cross-multiplying gives:
\[ B^3 = 112 \times 196 \]

### Step 5: Calculate \( B \)
Now we need to calculate \( 112 \times 196 \):
\[ 112 \times 196 = 112 \times (14 \times 14) = (112 \times 14) \times 14 \]
Calculating \( 112 \times 14 \):
\[ 112 \times 14 = 1568 \]
Thus:
\[ B^3 = 1568 \]

### Step 6: Factorize and Solve for B
We can factor \( 1568 \):
\[ 1568 = 14^3 \times 2^3 \]
Taking the cube root:
\[ B = 28 \]

### Step 7: Calculate A
Now substitute \( B \) back into equation (1) to find \( A \):
\[ A \times 28 = 196 \]
Thus:
\[ A = \frac{196}{28} = 7 \]

### Conclusion
The two numbers are:
\[ A = 7 \quad \text{and} \quad B = 28 \]