Using a Problem Solving Technique for a simple word problem

[cbarker1]

Summary:: I have a document which was written by previous tutor supervisor/instructor at my undergraduate school. I want to learn how to use this technique for other math/physics subject besides algebra... I will provide an example of how to use this technique. I found an old textbook (College Algebra by Richard Heineman) with word problem on exercise 15 problem 1.

Dear Everyone,

Disclaimer: I am not in school, or this is not a homework question for me. I am using an outdate math book (meaning no body in my knowledge is teaching out of this book) to learn this technique.

Background information:
The Technique is using some transition words to help with solving words problems. Those words are (Let, then, But, so, Therefore). I believe this technique is kind prototype thinking of proof-writing for the pre-proof-writing classes (mainly, Pre-Calculus and Calculus, and not Geometry). I believe this technique will help with me in other situation like in differential equations, calculus, and other applied math fields. There is an example to illustrate this technique in the document:
"Suppose in designing a house, the living room is to be thrice as long as it is wide. The total area of the room is 507 square feet. What should be the length of the room?"

Solution to this example:
Let $w$ be the width. Then the length is $3w$, and the area will be $(3w)(w)=3w^2$. But we know the total area is 507. So $3w^2=507$. After we divide 3 and take the positive root on both side of the equation, we discovered $w=13$. Therefore (or any synonym), the length of the room is 39 feet because $3 \cdot 13=39$.

This is the technique that I want to learn...

---

So here is my problem:

The difference of the squares of two consecutive integers is 43. Find the integers.

My work is the following:

"Let $x$ be an integer. Then a consecutive integer is #x+1#, and the difference of the square of two consecutive integers is $x^2-(x+1)^2$. But we know that the difference is 43. So $x^2-(x+1)^2=43$..."

I know the answer should be whatever $x$ is then $x+1$ will be next answer, also.
I think I have made a mistake on the algebraic expression; I don't know what is the error that I made.

Any help and/or suggestions will be appreciated.

Thanks,
cbarker1

 

[symbolipoint]

Summary:: I have a document which was written by previous tutor supervisor/instructor at my undergraduate school. I want to learn how to use this technique for other math/physics subject besides algebra... I will provide an example of how to use this technique. I found an old textbook (College Algebra by Richard Heineman) with word problem on exercise 15 problem 1.

"Suppose in designing a house, the living room is to be thrice as long as it is wide. The total area of the room is 507 square feet. What should be the length of the room?"

I am probably missing the point of what you are trying to do, but the exercise as written is a common type of exercise, and may be solvable using fairly common algebra or arithmetic skills.

w for width of the room.
3w for length of the room.
3w*w=507
3w^2=507
w^2=507/3
w^2=169
(w^2)^(1/2)=169^(1/2)
w=13, the width of the room in feet
So then length of room is 39 feet.

 

[cbarker1]

I am probably missing the point of what you are trying to do, but the exercise as written is a common type of exercise, and may be solvable using fairly common algebra or arithmetic skills.

w for width of the room.
3w for length of the room.
3w*w=507
3w^2=507
w^2=507/3
w^2=169
(w^2)^(1/2)=169^(1/2)
w=13, the width of the room in feet
So then length of room is 39 feet.

I think this was an example for me and the other tutors to follow these steps in order to help the intermediate algebra students when I was undergraduate school. I agree with you that this is a common algebra skills. I also struggles with word problems. And I want to remedy that struggle.

 

[hmmm27]

Works for me (difference of squares). Where are you going wrong ?

 

[cbarker1]

I got a negative number as my solution. The back of the book, its positive.

 

[symbolipoint]

Works for me (difference of squares). Where are you going wrong ?

How am I missing this? The given exercise is unrelated to any Difference OF Squares.

edit: Maybe I misunderstood that o.p. gave two separate exercises. I only recognized his first one.

 

[hmmm27]

I got a negative number as my solution. The back of the book, its positive.

To me (and apparently the author) the question is looking for an absolute value as an answer.

 

[symbolipoint]

Now I understand what your second exercise question means. Let me start it, do some of the steps, but not finish.

Two consecutive numbers, x and x+1, have difference of squares be 43.
(x+1)^2-x^2=43, the meaning of the description.

Factorize the left side.
((x+1)-x)((x+1)+x)=43

Simplify inside the outer parentheses.
(x+1-x)(x+1+x)=43
(1)(2x+1)=43
2x+1=43
and from here you can do the steps which clearly follow...

 

[Mark44]

The difference of the squares of two consecutive integers is 43. Find the integers.
My work is the following:

"Let $x$ be an integer. Then a consecutive integer is #x+1#, and the difference of the square of two consecutive integers is $x^2-(x+1)^2$. But we know that the difference is 43. So $x^2-(x+1)^2=43$..."

Edit:
What I wrote below is true for positive x and x + 1, but not true for negative values.
This was touched on without comment by @symbolipoint. The only way for the difference of the squares of x and x + 1 to be positive 43 is if you subtract the smaller square from the larger, not the other way around as you showed, above.

So the correct equation is $(x + 1)^2 - x^2 = 43$

The reason your (@cbarker1) equation is wrong is this:
x < x + 1, which is true for any real number x.
It follows that $x^2 < (x + 1)^2$, and that $x^2 - (x + 1)^2 < 0$. This means that the difference $x^2 - (x + 1)^2$ will always be negative, so can't possibly be equal to any positive number.

 

[cbarker1]

This was touched on without comment by @symbolipoint. The only way for the difference of the squares of x and x + 1 to be positive 43 is if you subtract the smaller square from the larger, not the other way around as you showed, above.

So the correct equation is $(x + 1)^2 - x^2 = 43$

The reason your (@cbarker1) equation is wrong is this:
x < x + 1, which is true for any real number x.
It follows that $x^2 < (x + 1)^2$, and that $x^2 - (x + 1)^2 < 0$. This means that the difference $x^2 - (x + 1)^2$ will always be negative, so can't possibly be equal to any positive number.

Thank you for your explanation of why I was wrong with my reasoning.

 

[kuruman]

$\dots$I believe this technique is kind prototype thinking of proof-writing for the pre-proof-writing classes (mainly, Pre-Calculus and Calculus, and not Geometry)$~\dots$

I was just curious about why are you chose to exclude Geometry. I think words like "Let", "Therefore" and "But" are useful in any intellectual endeavor where you have to line up and present your thoughts in a logical manner.

 

[gleem]

It follows that x2<(x+1)2, and that x2−(x+1)2<0. This means that the difference x2−(x+1)2 will always be negative, so can't possibly be equal to any positive number.

Actually, the OP solution of the problem is not invalid since it was not stated that the integers should be positive. The solution of x= -22 satisfies the goal of the problem i.e., two consecutive integers the difference of whose squares is equal to 43.

 

[Mark44]

Actually, the OP solution of the problem is not invalid since it was not stated that the integers should be positive. The solution of x= -22 satisfies the goal of the problem i.e., two consecutive integers the difference of whose squares is equal to 43.

None of the work I showed assumed that the integers had to be positive.

 

[vela]

Your conclusion, however, assumed $x \ge 0$. If $x$ is a negative integer, $(x+1)^2$ is less than $x^2$, so $x^2 - (x+1)^2$ will be positive. Try $x=-1$ for example.

 

[Mark44]

Your conclusion, however, assumed $x \ge 0$. If $x$ is a negative integer, $(x+1)^2$ is less than $x^2$, so $x^2 - (x+1)^2$ will be positive. Try $x=-1$ for example.

I stand corrected...

 

[hmmm27]

Since the equation works out as positive, how to denote that there's two solutions ?

 

[PeroK]

Since the equation works out as positive, how to denote that there's two solutions ?

The solutions are $-22, -21$ and $21, 22$.

 

[hmmm27]

The solutions are $-22, -21$ and $21, 22$.

Thankyou, but I meant how to write the equations such that it's obvious that there's 2 solutions. (or is it something too obvious).

 

[PeroK]

Thankyou, but I meant how to write the equations such that it's obvious that there's 2 solutions. (or is it something too obvious).

You mean $$(x+1)^2 - x^2 = \pm 43$$

 

[cbarker1]

I was just curious about why are you chose to exclude Geometry. I think words like "Let", "Therefore" and "But" are useful in any intellectual endeavor where you have to line up and present your thoughts in a logical manner.

I exclude geometry because it is a proof writing course before calculus. I agree with your assessment.

 

[Mark44]

The Technique is using some transition words to help with solving words problems. Those words are (Let, then, But, so, Therefore). I believe this technique is kind prototype thinking of proof-writing for the pre-proof-writing classes (mainly, Pre-Calculus and Calculus, and not Geometry).

I was just curious about why are you chose to exclude Geometry. I think words like "Let", "Therefore" and "But" are useful in any intellectual endeavor where you have to line up and present your thoughts in a logical manner.

I exclude geometry because it is a proof writing course before calculus. I agree with your assessment.

When I took geometry, many years ago, writing proofs with sequences of logically connected statements was a significant part of the class, so @kuruman's point is valid, at least insofar as high school geometry was taught back then. I say "back then" because many schools these days have chosen to mush together a lot of what used to be separate disciplines of algebra and geometry. I had Algebra I in 9th grade, and Geometry in 10th grade.

 

[gleem]

When I took geometry, many years ago, writing proofs with sequences of logically connected statements was a significant part of the class, so @kuruman's point is valid, at least insofar as high school geometry was taught back then. I say "back then" because many schools these days have chosen to mush together a lot of what used to be separate disciplines of algebra and geometry. I had Algebra I in 9th grade, and Geometry in 10th grade.

As I recall that was the "selling point" for my taking geometry (tenth grade too). In fact, the teacher made it a point to say that Abraham Lincoln studied geometry to improve his reasoning ability.

From: https://www.wsj.com/articles/what-honest-abe-learned-from-geometry-11621656062

In 1865, the Reverend J.P. Gulliver asked Abraham Lincoln how he came to acquire his famous rhetorical skill. The President gave an unusual response:

“In the course of my law-reading I constantly came upon the word ‘demonstrate.’ I thought, at first, that I understood its meaning, but soon became satisfied that I did not…. At last I said, ‘Lincoln, you can never make a lawyer if you do not understand what demonstrate means’; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any propositions in the six books of Euclid at sight.”

 

[kuruman]

When I took geometry, many years ago, writing proofs with sequences of logically connected statements was a significant part of the class, so @kuruman's point is valid, at least insofar as high school geometry was taught back then. I say "back then" because many schools these days have chosen to mush together a lot of what used to be separate disciplines of algebra and geometry. I had Algebra I in 9th grade, and Geometry in 10th grade.

I had three semesters of high school plane Euclidean geometry and one semester of solid geometry. Most certainly, that sequence taught me how to know when I have successfully "shown" that (or demonstrated as Abe Lincoln would say) something is true. I am chagrined to see that geometry is no longer treasured as a mental exercise.

 

[PeroK]

I had three semesters of high school plane Euclidean geometry and one semester of solid geometry. Most certainly, that sequence taught me how to know when I have successfully "shown" that (or demonstrated as Abe Lincoln would say) something is true. I am chagrined to see that geometry is no longer treasured as a mental exercise.

I didn't know you went to school with Lincoln!

 

[kuruman]

I don't remember him in my class. He must have sat in the back row and kept to himself. I do remember, however, a result that we proved (with compass and ruler of course): "The area of a triangle is equal to the product of the three sides divided by twice the diameter of the circumscribed circle." I don't remember anymore how to do it - maybe Abe does.