Thread: Problem Solving Using Algebraic Models

"The Chunnel connects the United Kingdom and France by a railway tunnel under the English Channel. The British started tunneling 2.5 months before the French and averaged 0.63 kilometers per month. The French averaged 0.47 kilometers per month. When the two sides met, they had tunneled 37.9 kilometers. How many kilometers did each country build? If the French started tunneling on February 28, 1988, approximately when did the two sides meet?"

And there's this one, if you feel like being extra helpful, too.

"Some of your classmates ask you to type their history papers throughout a 7 week summer course. How much should you charge per page, if you want to earn enough to pay for the flying lessons in Exercise 20 (The lessons are 80 dollars each and the introductory course is 5/8 of the normal lesson price. ) and have 75 dollars left over for spending money? You estimate you can type 40 pages per week. Assume that you have to take 9 flying lessons plus the introductory lesson and that you already have $375 to spend on lessons."

I've been working on it, but still haven't found the answer. The only way I could think of setting it up was like this:

1.575 + .63e + .47f = 37.9

With 1.575 being added for England's additional 2.5 months, the e being England and the f being France. But that would not work out without more information being given out. It just sounds like it won't work unless I know how many months the thing took to build or how much one of the countries contributed.

I only have one hour to solve five of these word problems, so any help would seriously be appreciated.

Originally Posted by Shelbyrockin

"The Chunnel connects the United Kingdom and France by a railway tunnel under the English Channel. The British started tunneling 2.5 months before the French and averaged 0.63 kilometers per month. The French averaged 0.47 kilometers per month. When the two sides met, they had tunneled 37.9 kilometers. How many kilometers did each country build? If the French started tunneling on February 28, 1988, approximately when did the two sides meet?"

And there's this one, if you feel like being extra helpful, too.

"Some of your classmates ask you to type their history papers throughout a 7 week summer course. How much should you charge per page, if you want to earn enough to pay for the flying lessons in Exercise 20 (The lessons are 80 dollars each and the introductory course is 5/8 of the normal lesson price. ) and have 75 dollars left over for spending money? You estimate you can type 40 pages per week. Assume that you have to take 9 flying lessons plus the introductory lesson and that you already have $375 to spend on lessons."

My style is I give complete answers. That is mostly because I want to avoid additional questions from the poster/asker. Also, related to that, I want to avoid back and forth asking and answering with the poster. I assume that my answers are detailed enough for the poster/asker to understand.
If, however, the poster still cannot follow my answer, then I welcome further questions from the poster for me to clarify anything he does not understand yet in my answer.

Solving word problems is easy as long as you know how to put into math, uh, "language" the wordings of the problem. First and formost is you must understand very well what the problem asks for. Then correlate that to the given facts in the problem.

Problem (1):
There are two things the problem asks you to find:
a)how many kilometers did each team build?
b)In what date did the two teams meet if the French started on 28Feb'88?

1.a)Given:
British....started 2.5 months earlier, and builds an average of 0.63 km per month.
French.....builds an average of 0.47 km per month.
When they met, they had built a total of 37.9 km.

Just like distance, task = rate * time, so when the two teams met,
British....has built [(0.63)(2.5) +(0.63)(t)]km ...where t is time in months.
French ....has built (0.47)(t) km.
So, for total distance,
(0.63)(2.5) +(0.63)(t) +(0.47)(t) = 37.9
1.575 +(1.10)(t) = 37.9
(1.1)t = 37.9 -1.575 = 36.325
t = 36.325 /1.1 = 33 months --------**
Meaning, 33 months after the French started tunneling, the two teams met and finished the tunneling.

Therefore,
British....built (0.63)(2.5) +(0.63)(33) = 22.37 km -----answer.
French.....built (0.47)(33) = 15.51 km -----------------answer.


1.b)
The two teams met in 33 months after 28Feb'88.
That was about on December 1, 1990. --------------answer.

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Problem 2);

The problem asks you how much you are going to charge per page so that you can take the lessons and you still have $75 for spending money.

How much do you need then?

You have initially $375.
You need to pay for the Introductotry lesson at (5/8)($80) and 9 Lessons at $80 each.
And you have to have $75 after paying for all those 10 lessons.
So,
75 +(5/8)(80) +9(80) -375 = $470 ----you must earn from your typing.

You can type 40 pages per week, and you have 7 weeks to type.
So if you charge x dollars per page,
(40*7)x = 470
x = 470/280
x = 1.67857 dollars per page.

You may round that off to $1.70 per page so thast you may not need to type the whole 7 weeks.
Or you can charge $2.00 per page if you want to relax a little in your typing.
But do not charge less than $1.68 per page.