The sum of diameter of two circles is 14 cm and the difference of their circumferences is 8 cm. Find the circumference of two circles

Answer

To solve the problem, we need to find the circumferences of two circles given the sum of their diameters and the difference of their circumferences.

### Step-by-Step Solution:

1. **Let the diameters of the circles be \(d_1\) and \(d_2\)**:
- According to the problem, we have:
\[ d_1 + d_2 = 14 \quad \text{(Equation 1)} \]

2. **Express the circumferences of the circles**:
- The circumference of a circle is given by the formula \(C = \pi d\). Therefore:
\[ C_1 = \pi d_1 \quad \text{and} \quad C_2 = \pi d_2 \]
- The problem states that the difference of their circumferences is 8 cm:
\[ C_1 - C_2 = 8 \quad \text{(Equation 2)} \]

3. **Substituting the circumference formulas into Equation 2**:
- We can substitute \(C_1\) and \(C_2\) from above:
\[ \pi d_1 - \pi d_2 = 8 \]
- Factoring out \(\pi\):
\[ \pi (d_1 - d_2) = 8 \]
- Dividing both sides by \(\pi\) (using \(\pi \approx \frac{22}{7}\)):
\[ d_1 - d_2 = \frac{8 \cdot 7}{22} = \frac{56}{22} = \frac{28}{11} \quad \text{(Equation 3)} \]

4. **Now we have two equations**:
- From Equation 1: \(d_1 + d_2 = 14\)
- From Equation 3: \(d_1 - d_2 = \frac{28}{11}\)

5. **Adding Equation 1 and Equation 3**:
\[ (d_1 + d_2) + (d_1 - d_2) = 14 + \frac{28}{11} \]
- This simplifies to:
\[ 2d_1 = 14 + \frac{28}{11} \]
- To add these, convert 14 into a fraction:
\[ 14 = \frac{154}{11} \]
- Therefore:
\[ 2d_1 = \frac{154}{11} + \frac{28}{11} = \frac{182}{11} \]
- Dividing by 2:
\[ d_1 = \frac{182}{22} = \frac{91}{11} \text{ cm} \]

6. **Finding \(d_2\)**:
- Substitute \(d_1\) back into Equation 1:
\[ d_2 = 14 - d_1 = 14 - \frac{91}{11} \]
- Convert 14 into a fraction:
\[ d_2 = \frac{154}{11} - \frac{91}{11} = \frac{63}{11} \text{ cm} \]

7. **Finding the circumferences**:
- Now we can find the circumferences \(C_1\) and \(C_2\):
\[ C_1 = \pi d_1 = \frac{22}{7} \cdot \frac{91}{11} = \frac{22 \cdot 91}{77} = \frac{2002}{77} = 26 \text{ cm} \]
\[ C_2 = \pi d_2 = \frac{22}{7} \cdot \frac{63}{11} = \frac{22 \cdot 63}{77} = \frac{1386}{77} = 18 \text{ cm} \]

### Final Answer:
- The circumferences of the two circles are:
- \(C_1 = 26 \text{ cm}\)
- \(C_2 = 18 \text{ cm}\)