Four points are on a line segment, as shown.

If $A B: B C=1: 2$ and $B C: C D=8: 5$, then $A B: B D$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-149.jpg?height=67&width=491&top_left_y=1365&top_left_x=1294)

equals
(A) $4: 13$
(B) $1: 13$
(C) $1: 7$
(D) $3: 13$
(E) $4: 17$
Four points are on a line segment, as shown.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-254.jpg?height=73&width=548&top_left_y=213&top_left_x=1363)

If $A B: B C=1: 2$ and $B C: C D=8: 5$, then $A B: B D$

equals
(A) 4:13
(B) $1: 13$
(D) 3:13
(E) 4:17
(C) $1: 7$

## Solution

In order to compare the given ratios, we must rewrite the ratio $A B: B C=1: 2$ as $A B: B C=4: 8$. Now both ratios express $B C$ as 8 units and we can write $A B: B C: C D=4: 8: 5$.

Thus, $A B: B D=4:(8+5)$

$$
=4: 13
$$

ANSWER: (A)