If `A : B : C = 2 : 3 : 5` and `C : D = 2:5`, what is `A : D`?

Options

A. `4 : 25`
B. `8:21`
C. `6 : 25`
D. `12:35`

Answer

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

### Step-by-Step Solution:

1. **Express the Ratios in Terms of a Common Variable:**
From the ratio \( A : B : C = 2 : 3 : 5 \), we can express the values of \( A \), \( B \), and \( C \) in terms of a variable \( k \):
\[ A = 2k, \quad B = 3k, \quad C = 5k \]

2. **Express \( D \) in Terms of \( C \):**
From the ratio \( C : D = 2 : 5 \), we can express \( D \) in terms of \( C \):
\[ C = 2m, \quad D = 5m \]
To relate \( C \) from the first ratio to \( C \) in the second ratio, we set:
\[ 5k = 2m \]
From this, we can solve for \( m \):
\[ m = \frac{5k}{2} \]

3. **Substitute \( m \) Back to Find \( D \):**
Now substituting \( m \) back into the expression for \( D \):
\[ D = 5m = 5 \left(\frac{5k}{2}\right) = \frac{25k}{2} \]

4. **Find the Ratio \( A : D \):**
Now we have \( A = 2k \) and \( D = \frac{25k}{2} \). The ratio \( A : D \) can be expressed as:
\[ A : D = 2k : \frac{25k}{2} \]
To simplify this, we can multiply both sides by 2 to eliminate the fraction:
\[ A : D = 4k : 25k \]
Since \( k \) is a common factor, it cancels out:
\[ A : D = 4 : 25 \]

### Final Answer:
Thus, the ratio \( A : D \) is \( 4 : 25 \).