If `A: B = 3 : 4, B : C = 5 : 7` and `C: D = 8:9` then `A: D` is equal to

Options

A. `3:7`
B. `7:3`
C. `21:10`
D. `10:21`

Answer

To solve the problem, we need to find the ratio \( A : D \) given the ratios \( A : B = 3 : 4 \), \( B : C = 5 : 7 \), and \( C : D = 8 : 9 \).

### Step-by-Step Solution:

1. **Express Ratios in Fraction Form**:
- From \( A : B = 3 : 4 \), we can express it as:
\[ \frac{A}{B} = \frac{3}{4} \]
- From \( B : C = 5 : 7 \), we can express it as:
\[ \frac{B}{C} = \frac{5}{7} \]
- From \( C : D = 8 : 9 \), we can express it as:
\[ \frac{C}{D} = \frac{8}{9} \]

2. **Find a Common Value for B**:
- To combine these ratios, we need to express \( B \) in a way that allows us to link all ratios. We can set \( B = 20 \) (the least common multiple of 4 and 5).
- Now, using \( B = 20 \):
- From \( A : B = 3 : 4 \):
\[ A = \frac{3}{4} \times 20 = 15 \]
- From \( B : C = 5 : 7 \):
\[ C = \frac{7}{5} \times 20 = 28 \]

3. **Find D Using C**:
- Now, using \( C = 28 \) in the ratio \( C : D = 8 : 9 \):
\[ D = \frac{9}{8} \times 28 = 31.5 \]

4. **Calculate the Ratio \( A : D \)**:
- Now, we have \( A = 15 \) and \( D = 31.5 \):
\[ A : D = 15 : 31.5 \]
- To simplify this ratio, we can multiply both sides by 2 to eliminate the decimal:
\[ A : D = 30 : 63 \]
- Now, we can simplify this ratio:
\[ A : D = \frac{30}{3} : \frac{63}{3} = 10 : 21 \]

### Final Answer:
Thus, the ratio \( A : D \) is equal to \( 10 : 21 \).