If `A : B = 2 : 3, B : C = 3 : 4 and C :D = 3 : 5` then value of `A : B : C : D` is :

Options

A. `6 : 9 : 15 : 20`
B. `2 : 3: 4: 5`
C. `2 : 3 : 5 : 12`
D. `6 : 9 : 12 : 20`

Answer

To solve the problem of finding the ratio \( A : B : C : D \) given the ratios \( A : B = 2 : 3 \), \( B : C = 3 : 4 \), and \( C : D = 3 : 5 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
- \( A : B = 2 : 3 \)
- \( B : C = 3 : 4 \)
- \( C : D = 3 : 5 \)

### Step 2: Express each variable in terms of a common variable
Let’s express \( A \), \( B \), \( C \), and \( D \) in terms of a common variable.

From \( A : B = 2 : 3 \):
- Let \( A = 2x \) and \( B = 3x \) for some variable \( x \).

From \( B : C = 3 : 4 \):
- Since \( B = 3x \), we can express \( C \) as:
\[ C = \frac{4}{3}B = \frac{4}{3}(3x) = 4x \]

From \( C : D = 3 : 5 \):
- Since \( C = 4x \), we can express \( D \) as:
\[ D = \frac{5}{3}C = \frac{5}{3}(4x) = \frac{20}{3}x \]

### Step 3: Write all variables together
Now we have:
- \( A = 2x \)
- \( B = 3x \)
- \( C = 4x \)
- \( D = \frac{20}{3}x \)

### Step 4: Find a common denominator
To express \( A : B : C : D \) in a simple form, we can eliminate the fraction by multiplying all terms by 3 (the denominator of \( D \)):
- \( A = 2x \times 3 = 6x \)
- \( B = 3x \times 3 = 9x \)
- \( C = 4x \times 3 = 12x \)
- \( D = \frac{20}{3}x \times 3 = 20x \)

### Step 5: Write the final ratio
Thus, the ratio \( A : B : C : D \) becomes:
\[ A : B : C : D = 6x : 9x : 12x : 20x \]
This simplifies to:
\[ 6 : 9 : 12 : 20 \]

### Step 6: Simplify the ratio
To simplify \( 6 : 9 : 12 : 20 \), we can divide each term by 3:
\[ 2 : 3 : 4 : \frac{20}{3} \]
However, since we want to keep it in whole numbers, we can multiply through by 3:
\[ 6 : 9 : 12 : 20 \]

### Final Answer
Thus, the final ratio \( A : B : C : D \) is:
\[ 6 : 9 : 12 : 20 \]