# Ratio problem.

• Aug 13th 2010, 10:02 PM
el33t
Ratio problem.
Ok, here is the problem.

Let A,B and C be three points in a plane such that AB:BC = 3:5. Which of the following can be the ratio AB:AC ?

I : 1:2
II : 1:3
III : 3:8

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II and III

• Aug 14th 2010, 03:59 AM
HallsofIvy
Because the ratio of AB to BC is 3 to 5, we can take our unit of measurement to such that AB is 3 and BC is 5. Let C be some other point in the plane and let "x" be the distance from A to C. C can lie anywhere on the circle of radius 5 with center B. That circle crosses the line through A and B twice- and those mark the shortest and longest possible distance from B to C. When the circle crosses the line on the other side of B from A, the distance from B to C is 5- 3= 2 and when it crosses the line on the other side of A from B, the distance from B to C is 5+ 3= 8. The ratio of AB to AC must be between 3/8 and 3/2.
• Aug 14th 2010, 04:31 AM
el33t
HallsofIvy:

Ok, I understood that the ratio must lie between 3/8 and 3/2. So, your choice is (E) ? Because all the three given ratios lies between 3/8 and 3/2.

But according to the book (BARRON'S SAT) from where I took this question, the answer is (D). I don't know how they got this. Or whether I'm doing some mistake in the middle.
• Aug 14th 2010, 12:42 PM
Soroban
Hello, el33t!

Quote:

Let $\displaystyle A,B,C$ be three points in a plane such that: .$\displaystyle AB:BC \:=\: 3:5$

Which of the following can be the ratio $\displaystyle AB:AC$ ?

. . $\displaystyle \text{(I) }\;1\!:\!2 \qquad \text{(II) }\;1\!:\!3 \qquad \text{(III) }\;3\!:\!8$

. . $\displaystyle \begin{array}{ccc}(A) & \text{I only} \\ (B) & \text{II only} \\ (C) & \text{III only} \\ (D)& \text{ I and III only} & \Leftarrow \\ (E) & \text{I, II and III} \end{array}$

$\displaystyle \text{(I)}\;1\!:\!2$ is possible.

The three points can be placed like this:

Code:

                        o C                     * *               6  *  *               *    * 5             *      *         *        *     A o  *  *  *  o B             3

$\displaystyle \text{(III)}\:3\!:\!5$ is possible.

The three points can be placed like this:

Code:

      : - - - - - 8 - - - - - :       o---------o-------------o       A  - 3 -  B  - - 5 - -  C

But $\displaystyle \text{(II)}\:1\!:\!3$ is not possible.

Code:

      : - - - - - - 9 - - - - - - : ?       o---------o-------------o       A  - 3 -  B  - - 5 - -  C
• Aug 14th 2010, 08:52 PM
el33t