If A:B = 3:4, B: C= 5:7 and C:D = 8: 9, then, the ratio A:D is :

Options

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

Answer

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

1. **Write down the given ratios:**
- A:B = 3:4
- B:C = 5:7
- C:D = 8:9

2. **Express each variable in terms of a common variable:**
- Let A = 3x, B = 4x (from A:B)
- From B:C, we have B = 5y and C = 7y. Since B = 4x, we can equate:
\[ 4x = 5y \implies y = \frac{4x}{5} \]
- Now substituting y into C:
\[ C = 7y = 7 \left(\frac{4x}{5}\right) = \frac{28x}{5} \]

3. **Now express D in terms of x:**
- From C:D, we have C = 8z and D = 9z. Since C = \(\frac{28x}{5}\), we can equate:
\[ \frac{28x}{5} = 8z \implies z = \frac{28x}{40} = \frac{7x}{10} \]
- Now substituting z into D:
\[ D = 9z = 9 \left(\frac{7x}{10}\right) = \frac{63x}{10} \]

4. **Now we have A and D in terms of x:**
- A = 3x
- D = \(\frac{63x}{10}\)

5. **Find the ratio A:D:**
\[ A:D = \frac{3x}{\frac{63x}{10}} = \frac{3x \cdot 10}{63x} = \frac{30}{63} \]

6. **Simplify the ratio:**
\[ \frac{30}{63} = \frac{10}{21} \]

Thus, the ratio A:D is **10:21**.