If `A:B=2:5,B:C=4:3andC:D=2:1`, what is the value of `A:C:D` ?

Options

A. `6:5:2`
B. `7:20:10`
C. `8:30:15`
D. `16:30:15`

Answer

To solve the problem, we need to find the combined ratio \( A:C:D \) given the ratios \( A:B = 2:5 \), \( B:C = 4:3 \), and \( C:D = 2:1 \).

### Step-by-Step Solution:

1. **Write the Ratios**:
- From the problem, we have:
- \( A:B = 2:5 \)
- \( B:C = 4:3 \)
- \( C:D = 2:1 \)

2. **Express Ratios in Fraction Form**:
- We can express these ratios as fractions:
- \( \frac{A}{B} = \frac{2}{5} \)
- \( \frac{B}{C} = \frac{4}{3} \)
- \( \frac{C}{D} = \frac{2}{1} \)

3. **Find \( A:C \)**:
- To find \( A:C \), we can multiply the ratios \( A:B \) and \( B:C \):
\[ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} = \frac{2}{5} \times \frac{4}{3} \]
- Performing the multiplication:
\[ \frac{A}{C} = \frac{2 \times 4}{5 \times 3} = \frac{8}{15} \]
- So, \( A:C = 8:15 \).

4. **Find \( C:D \)**:
- We already have \( C:D = 2:1 \).

5. **Equalize the Value of \( C \)**:
- We need to express both ratios in terms of the same value for \( C \). The current value of \( C \) in \( A:C \) is 15, and in \( C:D \) it is 2.
- To equalize \( C \), we can multiply the ratio \( C:D \) by \( 15/2 \):
\[ C:D = 2 \times \frac{15}{2} : 1 \times \frac{15}{2} = 15:7.5 \]
- However, to keep it in whole numbers, we can multiply \( C:D \) by 15:
\[ C:D = 2 \times 15 : 1 \times 15 = 30:15 \]

6. **Combine Ratios**:
- Now we have:
- \( A:C = 8:15 \)
- \( C:D = 30:15 \)
- We can combine these to find \( A:C:D \):
- Since \( C \) is the same in both ratios, we can write:
\[ A:C:D = 8:15:30 \]

### Final Answer:
Thus, the value of \( A:C:D \) is \( 8:15:30 \).