If `A : B = (1)/(2) : (3)/(8) B : C = (1)/(3) : (5)/(9) and C : D = (5)/(6) : (3)/(4)` then the ratio A : B : C : D is

Options

A. A)`6 : 4 : 8 : 10`
B. B)` 6 : 8 : 9 : 10`
C. C)` 8 : 6 : 10 : 9`
D. D)` 4 : 6 : 8 : 10`

Answer

To solve the problem of finding the ratio A : B : C : D given the ratios A : B, B : C, and C : D, we can follow these steps:

### Step 1: Write down the given ratios
We have:
1. A : B = (1/2) : (3/8)
2. B : C = (1/3) : (5/9)
3. C : D = (5/6) : (3/4)

### Step 2: Simplify A : B
To simplify A : B = (1/2) : (3/8), we can cross-multiply:
- A/B = (1/2) / (3/8) = (1 * 8) / (2 * 3) = 8/6 = 4/3

So, A : B = 4 : 3.

### Step 3: Simplify B : C
Next, we simplify B : C = (1/3) : (5/9):
- B/C = (1/3) / (5/9) = (1 * 9) / (3 * 5) = 9/15 = 3/5

So, B : C = 3 : 5.

### Step 4: Simplify C : D
Now, we simplify C : D = (5/6) : (3/4):
- C/D = (5/6) / (3/4) = (5 * 4) / (6 * 3) = 20/18 = 10/9

So, C : D = 10 : 9.

### Step 5: Combine the ratios
Now we have:
- A : B = 4 : 3
- B : C = 3 : 5
- C : D = 10 : 9

To combine these ratios into A : B : C : D, we need to express them with a common term for B and C.

1. From A : B = 4 : 3, we can express B as 3k (where k is a common multiplier).
2. From B : C = 3 : 5, we can express C as (5/3) * B = (5/3) * 3k = 5k.
3. From C : D = 10 : 9, we can express D as (9/10) * C = (9/10) * 5k = (9 * 5) / 10 = 4.5k.

### Step 6: Write the combined ratios
Now we can write:
- A = 4k
- B = 3k
- C = 5k
- D = 4.5k

### Step 7: Express in whole numbers
To express these ratios in whole numbers, we can multiply each term by 2 to eliminate the decimal:
- A = 8k
- B = 6k
- C = 10k
- D = 9k

Thus, the combined ratio A : B : C : D = 8 : 6 : 10 : 9.

### Final Answer
The ratio A : B : C : D is 8 : 6 : 10 : 9.