If A : B = 4 : 5 , B : C = 3 : 7 , C : D = 6 : 7 D : E = 12 : 17
then find the value of ratio A : E.

Answer

To find the ratio A : E given the ratios A : B, B : C, C : D, and D : E, we can follow these steps:

### Step-by-Step Solution:

1. **Write down the given ratios:**
- A : B = 4 : 5
- B : C = 3 : 7
- C : D = 6 : 7
- D : E = 12 : 17

2. **Convert the ratios to fractions:**
- A/B = 4/5
- B/C = 3/7
- C/D = 6/7
- D/E = 12/17

3. **Express A in terms of E:**
To find the ratio A : E, we can multiply the fractions:
\[ A/E = (A/B) \times (B/C) \times (C/D) \times (D/E) \]

4. **Substitute the fractions:**
\[ A/E = \left(\frac{4}{5}\right) \times \left(\frac{3}{7}\right) \times \left(\frac{6}{7}\right) \times \left(\frac{12}{17}\right) \]

5. **Multiply the numerators and denominators:**
- Numerator: \( 4 \times 3 \times 6 \times 12 \)
- Denominator: \( 5 \times 7 \times 7 \times 17 \)

6. **Calculate the numerator:**
\[ 4 \times 3 = 12 \]
\[ 12 \times 6 = 72 \]
\[ 72 \times 12 = 864 \]

7. **Calculate the denominator:**
\[ 5 \times 7 = 35 \]
\[ 35 \times 7 = 245 \]
\[ 245 \times 17 = 4165 \]

8. **Combine the results:**
\[ A/E = \frac{864}{4165} \]

9. **Write the final ratio:**
Thus, the ratio A : E is:
\[ A : E = 864 : 4165 \]