The two extermes of a proportion are 12 and 14 and one of the middle terms is 24. Find the other middle term.

Answer

To solve the problem, we need to find the other middle term in the proportion where the two extremes are 12 and 14, and one of the middle terms is 24.

### Step-by-Step Solution:

1. **Identify the terms of the proportion**:
- The two extremes (first and fourth terms) are given as:
- First term (a) = 12
- Fourth term (d) = 14
- The middle terms (second and third terms) are represented as:
- Second term (b) = 24 (given)
- Third term (c) = ? (to be found)

2. **Set up the proportion**:
- The relationship of the terms in a proportion can be expressed as:
\[ \frac{a}{b} = \frac{c}{d} \]
- Substituting the known values into the equation:
\[ \frac{12}{24} = \frac{c}{14} \]

3. **Cross-multiply to solve for the unknown**:
- Cross-multiplying gives us:
\[ 12 \times 14 = 24 \times c \]
- This simplifies to:
\[ 168 = 24c \]

4. **Isolate the variable (c)**:
- To find c, divide both sides by 24:
\[ c = \frac{168}{24} \]

5. **Calculate the value of c**:
- Performing the division:
\[ c = 7 \]

6. **Conclusion**:
- The other middle term (c) is 7. Therefore, the two middle terms are 24 and 7.

### Final Answer:
The middle terms are 24 and 7.