Two people, $A$ and $B$, walk from P to Q and back. $A$ starts 1 hour after $B$, overtakes him 2 miles from Q, meets him 32 minutes later, and arrives at P when $B$ is 4 miles off. Find the distance from P to Q.
I really cannot think of anything to solve this problem. I tried drawing a diagram to model the situation, but that didn't seem to help very much. The problem I am having is translating the English to mathematical language; which is really what makes me struggle with word problems in general.
Even if I could just have the equations to describe the problem, that would help tremendously; my problem is forming the equations, but solving them is easy for me. If I could also get some tips for how to form equations based on the word problems, that would be much help. Thanks.
Best Answer
Let $d$ be the distance from P to Q. Let $r_a$ and $r_b$ be the walking rates of A and B respectively (in miles per hour).
The key facts are:
(1) B takes one hour longer than A to walk $d-2$ miles.
(2) In 32 minutes, the sum total distance walked by A and B is $4$ miles.
(3) B takes one hour longer to walk $2d-4$ miles than it takes A to walk $2d$ miles.
From (1) you get (4): $\frac{d-2}{r_b}=\frac{d-2}{r_a}+1$.
From (2) you get (5): $\frac{32}{60}\cdot(r_a+r_b)=4$
From (3) you get (6): $\frac{2d-4}{r_b}=\frac{2d}{r_a}+1$
This gives you three equations in three unknowns for you to solve, which you said would be the easy part for you. [It may be helpful to note that the left side of equation (6) is exactly twice the left side of equation (4).]