You are quite welcome! I'm glad you got it worked out. To work algebra word problems I usually
start by guessing at the answer. Then in working to see if it's the right guess I keep track step by
step as to how I determine whether it is correct or not. There is usually a step where something is
supposed to equal something else. If it does then the guess was correct. If not then putting a
brand x generic guess where the original numeric guess was and going through the same steps,
the equation to be solved shows up where the "they were not equal" occurred.
There are several advantages to this approach. 1) Anyone can make a guess and so can get
started on working the problem. 2) The guess doesn't usually even have to be reasonable.
3) After the guess, there is no information left out so one can reason through the problem in
the manner they are used to dealing with this type of problem. 4) If one can't determine whether
or not the guess is correct they they probably don't understand the background material or the
problem is not well stated. 5) If one does determine that the guess is correct then they have
the answer. 6) If the guess is incorrect then they are very close to having the equation to be
solved (equality involving the generic guess x following steps for guesses) . 7) Sometimes
after finding that a guess is wrong it is easy to adjust the guess to get the right answer.
The main advantage is probably that one is able to work out the steps in the "usual" manner
instead of having to work in a strange direction dictated by the "missing part" being looked for.
The "usual" manner leads to the equation to be solved. Then solving the equation is straight
forward which alleviates one from having to work through the problem from a "strange"
This approach was "driven home" to me when I ran across a "heads of lettuce" problem in an
algebra book. I had never seen anything like it before and so had no "standard approach" to
work with. So I took a guess, worked through it (it was a wrong guess) and tried several
revised guesses. They didn't work either, but I knew I was reasoning through the problem
correctly. So putting the "generic" guess x in instead of the specific guesses and going through
the SAME steps, the equation to be solved popped up. Solving it was a breeze. That made me
acutely aware of the method mentioned.
An in case you wonder whether guessing is a "valid" approach in mathematics, just know that
there are many places in mathematics that the "official" technique for solving (differential
equations, for example) is to "guess and correct, estimate and revise" or whatever they may
wish to call it. It is probably the most common way that human beings learn things. We guess
and if it doesn't work we guess and try again. We learned to crawl, walk, ride a bike, drive a car,
etc. by this approach. Much of mathematics is obtained by observing patterns and then guessing
what equation for formula "fits" the patterns.
Well, I didn't intend to write a book here. I get a bit verbose at times. But I hope this will be of
help to you.
Have a super day today!