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_ac36362783317e0251fdg-148.jpg?height=76&width=504&top_left_y=428&top_left_x=1279)
equals
(A) $4: 13$
(B) $1: 13$
(C) $1: 7$
(D) $3: 13$
(E) $4: 17$

## Part B: Each question is worth 6 credits.
Four points are on a line segment, as shown.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-302.jpg?height=76&width=566&top_left_y=2007&top_left_x=1297)

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)

## PART B: