SOLUTION: Points A and B are both on the line segment PQ on the same side of its midpoint. A divides PQ in the ratio 4:5, and B divides PQ in the ratio of 3:7. If AB=65, then what is the l

Algebra ->  Customizable Word Problem Solvers  -> Misc -> SOLUTION: Points A and B are both on the line segment PQ on the same side of its midpoint. A divides PQ in the ratio 4:5, and B divides PQ in the ratio of 3:7. If AB=65, then what is the l      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1123514: Points A and B are both on the line segment PQ on the same side of its midpoint. A divides PQ in the ratio 4:5, and B divides PQ in the ratio of 3:7. If AB=65, then what is the length of PQ?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
crazy problem, but this is how i got my answer.

hopefully it's right.

you are given that AB = 65

you are given A divides PQ in segments of 4x and 5x, where PQ is equal to 9x

you are given B divides PQ in segments of 3y and 7y, where PQ is equal to 10y

so you have 2 equations.

they are 4x + 5x = 9x and 3y + 7y = 10y

PQ is the same length, so you know that 9x = 10y

solve for y to get y = 9/10 * x

in the equation of 3y + 7y = 10y, replace y with 9/10 * x to get::

3 * 9/10 * x + 7 * 9/10 * x = 10 * 9/10 * x

this results in:

27/10 * x + 63/10 * x = 90/10 * x

multiply both sides of this equation by 10 to get:

27x + 63x = 90x

in the equation of 4x + 5x = 9x, multiply both sides of this equation by 10 to get:

40x + 50x = 90x

the segment PA is equal to 40x/90x of the total length of PQ.

the segment PB is equal to 27x/90x of the total length of PQ.

the segment AB is equal to the segment PA minus the segment PB.

the segment AB is equal to 65.

you get PA minus PB = 40x/90x - 27x/90x = 13x/90x = 13/90 of the length of PQ.

since 65 is the length of PA minus PB, then 13/90 * the length of PQ is equal to 65.

solve for the length of PQ to get PQ = 90/13 * 65 = 450.

the fact that A and B are on the same side of the midpoint of PQ is irrelevant to the problem as far as i can tell.

the solution is that the length of PQ is 450 units, whatever they are.

i have no idea if this is the preferred way to solve this, but i think it works.

i solved for the length of all segments and they do confirm the solution is correct.

PQ is 450
PA is 200
PB is 135
AQ is 250
BQ is 315
AB is PA minus PB = 200 - 135 = 65

i looked at a different way to solve it see if it might be easier.

i got:

PA = 4/9 * PQ
PB = 3/10 * PQ
AB = PA minus PB = 4/9 * PQ minus 3/10 * PQ
place these under a common denominator of 90 to get:
AB = 40/90 * PQ minus 27/90 * PQ = 13/90 * PQ

since AB = 65, you get 65 = 13/90 * PQ

solve for PQ to get PQ = 65 * 90/13 = 450

same answer, but possibly easier to understand.

in either case, the length of PQ looks like it's 450.