Book divided inot Chapters

J

Jim77

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Hello All

Can anyone help get me started on the following word problem:

"A book has three chapters called Audit, Business and Computing. The total number of pages in the book is 1 to 500 inclusive. Each chapters starts on a new page. The sum of the pages for the chapter Audit is double that of the sum of the pages for Business. The number of pages for the chapter Business is greater than twice that of the number of the pages for Computing.

How many pages in the Computing chapter?"

Thanks in advance.

Regards

Jim
 
Jim77 said:
Hello All

Can anyone help get me started on the following word problem:

"A book has three chapters called Audit, Business and Computing. The total number of pages in the book is 1 to 500 inclusive. Each chapters starts on a new page. The sum of the pages for the chapter Audit is double that of the sum of the pages for Business. The number of pages for the chapter Business is greater than twice that of the number of the pages for Computing.

How many pages in the Computing chapter?"

Thanks in advance.

Regards

Jim
Click to expand...

Start with

# of pages in Audit = A

# of pages in Business = B

# of pages in Computing = C

Given:

A + B + C = 500 .................................................(1)

continue...

Please share your work with us.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...217#post322217

We can help - we only help after you have shown your work - or ask a specific question (e.g. "are these correct?")
 
Thanks for your reply.

Sorry - really not sure where you are going with this one?
 
Jim77 said:
Thanks for your reply.

Sorry - really not sure where you are going with this one?
Click to expand...
Subhotosh Khan has done the first step in every word problem: assign letters to the unknowns in writing.

He has started the second step in every word problem: express the information in the problem in mathematical form using the letters assigned in the first step. (Sometimes you may need to supplement this with some general information not explicitly given in the problem.)

He is asking you to complete the second step, at which point you will have translated the word problem into a pure math problem..

The third step is to work out the math problem.

The fourth step is to check your answer.

Going back to the second step, there were three unknowns identified in the first step. To find three unknowns requires three equations. Subhotosh Khan has given you one of them. You need to find the other two.

What do you think they are?
 
Ok Jeff thanks for the steer.

Sum of the pages in A = 2*sum of the pages for B (2)
Number of pages for B > 2*the number of pages for C (3)

Am I on the right track?
 
Jim77 said:
Ok Jeff thanks for the steer.

Sum of the pages in A = 2*sum of the pages for B (2)
Number of pages for B > 2*the number of pages for C (3)

Am I on the right track?
Click to expand...
Very very close, but not quite. The whole point of using letters is so you can put things into mathematical language.

A = the sum of the pages in Audit

B = the sum of the pages in Business

C = the sum of the pages in Computing.

This is what SK did for the first step except he used the symbol for number instead of the words "the sum."

Now put the information in the problem into math speak. NO ENGLISH. You are translating from English into math speak in the second step.

SK gave you the first equation A + B + C = 500.

A = 2B. You got that equation right, but you kept it partially in English.

B > 2C. You got that inequation right, but you kept it partially in English.

That almost completes step 2, but you do not have three EQUATIONS. Is there any other piece of information in the problem to help find the answer?
 
Sorry Jeff I can't see any more information in the word problem we can use...

I have spotted an error. The sum of the pages for A, B and C does not equal 500. For example if the first chapter runs from page 1 to page 10 then the sum is 1+2+3+4+5+6+7+8+9+10 = 55
 
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Jim77 said:
Not so clever now are we? LOL
Click to expand...
No I misread the problem.

But still there is additional information implied in the problem. Each chapter starts on a different page. All the unknowns are positive integers.

\(\displaystyle A = \text{number of pages in Audit Chapter}.\)

\(\displaystyle B = \text{number of pages in Business Chapter.}\)

\(\displaystyle C = \text{number of pages in Computing Chapter.}\)

\(\displaystyle X = \text{the sum of the page numbers in Audit Chapter.}\)

\(\displaystyle Y = \text{the sum of the page numbers in Business Chapter.}\)

\(\displaystyle m = \text{page number of first page of second chapter.}\)

\(\displaystyle n = \text{page number of first page of third chapter.}\)

\(\displaystyle A,\ B,\ C,\ X,\ Y,\ m,\ and\ n\ \in \mathbb I^+.\)

\(\displaystyle A + B + C = 500 \implies 499 < A + B + C < 501.\)

\(\displaystyle X = 2Y.\)

\(\displaystyle B > 2C.\)

\(\displaystyle n \ge m + 1.\)

From there I need to think a bit.

By the way, it helps us to know what you are studying. I thought at first that you were studying elementary algebra. Seems I was wrong.
 
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First thought: does the problem specify the order of the chapters? That is, does Audit come before Business or not, or is that left open?
 
Jim77 said:
Hello All

Can anyone help get me started on the following word problem:

"A book has three chapters called Audit, Business and Computing. The total number of pages in the book is 1 to 500 inclusive. Each chapters starts on a new page. The sum of the pages for the chapter Audit is double that of the sum of the pages for Business. The number of pages for the chapter Business is greater than twice that of the number of the pages for Computing.

How many pages in the Computing chapter?"

Thanks in advance.

Regards

Jim
Click to expand...

The problem deals with the sum of the pages

NOT

the sum of the page-numbers
 
I believe the correct interpretation of the intention of the problem would be:

"A book has three chapters called Audit, Business and Computing. The total number of pages in the book is 1 to 500 inclusive. Each chapters starts on a new page. The sum of the number of pages for the chapter Audit is double that of the sum of the number of pages for Business. The number of pages for the chapter Business is greater than twice that of the number of the pages for Computing.

How many pages in the Computing chapter?"

If that is the case - then this problem would have a range (of answer)

First we see that 3B + C = 500 → C≥2

then we see that 6C + C < 500 → C < 71

We have 3B + C = 500 → C ≤ 68

Then 2 ≤ C ≤ 68
 
Thanks Jeff - I think there is no stipulation regarding the order of the chapters provided they obey the criteria.
 
Subhotosh Khan said:
I believe the correct interpretation of the intention of the problem would be:

"A book has three chapters called Audit, Business and Computing. The total number of pages in the book is 1 to 500 inclusive. Each chapters starts on a new page. The sum of the number of pages for the chapter Audit is double that of the sum of the number of pages for Business. The number of pages for the chapter Business is greater than twice that of the number of the pages for Computing.

How many pages in the Computing chapter?"

If that is the case - then this problem would have a range (of answer)

First we see that 3B + C = 500 → C≥2

then we see that 6C + C < 500 → C < 71

We have 3B + C = 500 → C ≤ 68

Then 2 ≤ C ≤ 68
Click to expand...

Sorry SK but the intention is definitely for the sum of the page numbers....
 
Subhotosh Khan said:
We have prospered in life and we are content → We don't need to prove that we are clever

However you do and remember that......
Click to expand...

I was only teasing and did not mean to offend...

I think you are correct perhaps I mis-wrote the original problem...It should have read the sum of the page numbers...
 
We need denis. I ask again what you are studying because I at least do not see a way to solve this using elementary algebra. It can be solved through the brute force of programming. A fairly simple program will find the answer or answers reasonably quickly. The reason is that A, B, and C are all positive integers less than 500. A loop inside a loop should find all the possible answers (if there are any).

It may be that some branch of mathematics that I do not know (Diophantine equations for example) can make quick work of this problem, but programming will certainly work.
 
Jim77 said:
The total number of pages in the book is 1 to 500 inclusive.
Click to expand...
Jim, that is not a logical statement.
One number would be the right of the word "is."
"The total number of pages in the book is 500." \(\displaystyle \ \ \ \ \)Look at the inconsistency of the words/phrases
in the problem statement:

" . . . Each chapters starts on a new page. The > > sum of the pages < < for the chapter Audit is double that of
the > > sum of the pages for Business. The > > number of pages [< < for the chapter Business is greater than
twice that of the > > number of the pages < < for Computing.

How many pages in the Computing chapter?"

_______________________________________________________________________________


Here is the meaning I get of what I believe the problem intends: "Each chapters starts on a new page.
The number of the pages for the chapter Audit is double that of the the number of the pages for Business.
The number of pages for the chapter Business is greater than twice that of the number of the pages for Computing.

How many pages in the Computing chapter?"
 
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