Points P and Q are both in the line segment AB and on the same side of its midpoint. P divides AB in the ratio 2 : 3, and Q divides AB in the ratio 3 : 4. If PQ = 2, then find the length of the line segment AB.
Draw a horizontal line. Label a point near the left-hand end as "A"; label a point near the right-hand end as "B".
You are given that P splits AB into parts in the ratio 2:3, so there are "five parts", of which two are to the left of P and three are to the right of P.
You are given that Q splits AB into parts in the ratio 3:4, so there are "seven parts", of which three are to the left of Q and four are to the right of Q.
To convert these into comparable "parts", let's split each of the "parts" related to P into seven smaller pieces, so there fourteen pieces to the left of P, and twenty-one pieces to the right of P. And let's split each of the "parts" related to Q into five smaller pieces, so there are fifteen pieces to the left of Q, and twenty pieces to the right of Q. In either case, we have split the line AB into thirty-five pieces, which then must be of equal size (which we can label as "p").
Given the relative numbers of pieces, which of P and Q must be to the left of the other? (That is, what letters should replace the query marks in the following, and in what order?)
Code:
line:
(15p) ? ? (20p)
---*-------*-*----------*---
A (14p) | | (21p) B
| |
|2|
So where does the segment PQ fall? How then does this segment's length fit in with the other information? Where does this lead?
If you get stuck, please reply showing your work so far. Thank you!