Let A,B,C,D be collinear points in that order. Suppose `AB: CD = 3:2 and BC : AD = 1:5`. Then `AC :BD` is(A) `1:1` (B)` 11:10` (C) `16:11` (D) ` 17:13`

Options

A. `1 : 1`
B. `11 : 10`
C. `16 : 11`
D. `17 : 13`

Answer

To solve the problem, we need to find the ratio \( AC : BD \) given the ratios \( AB : CD = 3 : 2 \) and \( BC : AD = 1 : 5 \).

### Step-by-Step Solution:

1. **Assign Variables Based on Ratios**:
- Let \( AB = 3x \) and \( CD = 2x \) for some positive value \( x \).
- Let \( BC = y \) and \( AD = 5y \) for some positive value \( y \).

2. **Express Total Lengths**:
- Since points A, B, C, and D are collinear, we can express the total lengths:
- \( AC = AB + BC = 3x + y \)
- \( BD = BC + CD = y + 2x \)

3. **Relate the Lengths**:
- We know that \( AD = AB + BC + CD \). Therefore, we can set up the equation:
\[ AD = AB + BC + CD \implies 5y = 3x + y + 2x \]
- Simplifying this gives:
\[ 5y = 5x + y \implies 5y - y = 5x \implies 4y = 5x \implies y = \frac{5}{4}x \]

4. **Substitute \( y \) in Terms of \( x \)**:
- Now substitute \( y \) back into the expressions for \( AC \) and \( BD \):
- \( AC = 3x + y = 3x + \frac{5}{4}x = \frac{12}{4}x + \frac{5}{4}x = \frac{17}{4}x \)
- \( BD = y + 2x = \frac{5}{4}x + 2x = \frac{5}{4}x + \frac{8}{4}x = \frac{13}{4}x \)

5. **Find the Ratio \( AC : BD \)**:
- Now we can find the ratio:
\[ AC : BD = \frac{17}{4}x : \frac{13}{4}x \]
- The \( x \) and \( \frac{1}{4} \) cancel out, giving:
\[ AC : BD = 17 : 13 \]

### Final Answer:
Thus, the ratio \( AC : BD \) is \( 17 : 13 \). The correct option is (D) \( 17 : 13 \).