I feel bad I can't understand the most simple of algebra

[BadFish]

Homework Statement

First, let me tell you a bit about myself. I did not do well in geometry and math in middle school, I was terrible at it. At high school, something changed and I took Algebra I. I was getting B's to high A's. I got awarded the rotary club award for student of the month, because of my grades at the time. I had a lot of problems in school with learning problems, so I decided to graduate from a charter school. I graduated and have been taking a long break, many years, I'd say at least 4 or 5 persuing other interests. I recently started some training in basic computer skills such as excel, powerpoint and word and am starting my internship this monday, 10 weeks, 2 hours a day, 2 days a week. After that I plan to get a job with basic clerical computer work, and study and go to college on the side with a pale grant. I have always enjoyed math immensly, and I got the book "calculus for dummies" and read it from cover to cover. The main thing was I could not do most of it like basic limits because I never learned how to factor.

I also skipped geometry and trigonmetry, as I just wanted to read out of pleasure. I enjoy reading about mathematics and solving problems. Recently when I was at computer school I picked up a GED book on factoring, hoping to finally learn it. I reviewed some algebra I hadn't done for years, and managed to get the factoring questions right. Needless to say I was happy with that. The main thing, I found out, was to find the greatest common factor of the terms and put that on the outside of the formula, then ask myself that number times "what" equals the outcome.

I do want to take some schooling in mathematics and perhaps engineering after I get really good. But I find myself not understand even the basic problems. I have a very good book (at least I think so) on algebra, it is called "Algebra, The easy way" and is presented in a story, which I find easier to learn on the most part, because it explains WHY things happen the way they do. However, I'm stuck on the first chapter! Here's the problem:

Verify the associative properties of addition and multiplication in the following cases. To do this, calculate the result of the expression in two different ways. (It does not say how)
For example,

3 x (4 x 5)= 3 x 20 = 60
(3 x 4) x 5= 12 x 5 = 60

Homework Equations

Problem 1:
12 x 6 x 2

Problem 2:
11 x 5 x 16

Part 3:
In excerises 20-31, write formulas that perform the indicated functions. (In each case you will need to think of the appropriate letters to represent the quantities indicated)
20. Calculate your pay if you work h hours in one week and are paid $5 per hour.

The Attempt at a Solution

For the first 2 problems, I'm really at a lost here. I learned how to factor polynomials with algebra, but I can't figure out this! It must be really simple. If someone could please give me a tip or hint or how to solve them, it would be greatly appreciated!

On part 3, I used to be good at word problems, I guess I can try to solve this one!
Let h equal the number of hours worked in one week. Then,
$5/per hour x 7h

Is this correct?

Thanks for helping me out, or giving me hints! I love this book and want to try and get back into math.

 

[Ouabache]

First, try and choose a category that "most closely fits" your question. In this case, you might chose Topic - Homework & Coursework Questions> Precalculus Mathematics.

For problems (1) and (2), in words, what is the associative property of multiplication? If you don't understand your book's definition, it is fine to look this up on the web. Use a search engine of your choice, and try some key words like: "associative property", math.
Then come back here and let us know what you learned.

For (3), what is the result you are trying to determine by your equation? (assign that a variable name). What other pieces of information have they given you?
Let us know what you've come up with and we can point you in a successful direction.

 

[HallsofIvy]

Homework Statement

First, let me tell you a bit about myself. I did not do well in geometry and math in middle school, I was terrible at it. At high school, something changed and I took Algebra I. I was getting B's to high A's. I got awarded the rotary club award for student of the month, because of my grades at the time. I had a lot of problems in school with learning problems, so I decided to graduate from a charter school. I graduated and have been taking a long break, many years, I'd say at least 4 or 5 persuing other interests. I recently started some training in basic computer skills such as excel, powerpoint and word and am starting my internship this monday, 10 weeks, 2 hours a day, 2 days a week. After that I plan to get a job with basic clerical computer work, and study and go to college on the side with a pale grant. I have always enjoyed math immensly, and I got the book "calculus for dummies" and read it from cover to cover. The main thing was I could not do most of it like basic limits because I never learned how to factor.

I also skipped geometry and trigonmetry, as I just wanted to read out of pleasure. I enjoy reading about mathematics and solving problems. Recently when I was at computer school I picked up a GED book on factoring, hoping to finally learn it. I reviewed some algebra I hadn't done for years, and managed to get the factoring questions right. Needless to say I was happy with that. The main thing, I found out, was to find the greatest common factor of the terms and put that on the outside of the formula, then ask myself that number times "what" equals the outcome.

I do want to take some schooling in mathematics and perhaps engineering after I get really good. But I find myself not understand even the basic problems. I have a very good book (at least I think so) on algebra, it is called "Algebra, The easy way" and is presented in a story, which I find easier to learn on the most part, because it explains WHY things happen the way they do. However, I'm stuck on the first chapter! Here's the problem:

Verify the associative properties of addition and multiplication in the following cases. To do this, calculate the result of the expression in two different ways. (It does not say how)
For example,

3 x (4 x 5)= 3 x 20 = 60
(3 x 4) x 5= 12 x 5 = 60

Homework Equations

Problem 1:
12 x 6 x 2

Problem 2:
11 x 5 x 16

Part 3:
In excerises 20-31, write formulas that perform the indicated functions. (In each case you will need to think of the appropriate letters to represent the quantities indicated)
20. Calculate your pay if you work h hours in one week and are paid $5 per hour.

The Attempt at a Solution

For the first 2 problems, I'm really at a lost here. I learned how to factor polynomials with algebra, but I can't figure out this! It must be really simple. If someone could please give me a tip or hint or how to solve them, it would be greatly appreciated!

Just follow the pattern shown in the first example.
Calculate (12 x 6) x 2 and 12 x (6 x 2). What do you get?

On part 3, I used to be good at word problems, I guess I can try to solve this one!
Let h equal the number of hours worked in one week. Then,
$5/per hour x 7h

Is this correct?

Where did the 7 come from?

Thanks for helping me out, or giving me hints! I love this book and want to try and get back into math.

 

[BadFish]

The associative property of multiplication means it doesn't matter where you put the parentheses, the product will be the same.

I don't understand how to rearrange the number like it does in the example

I think I am over thinking it.

As regards to the algebra problem, I guess I need a refresher on word problems. I used 7 because of 7 days in a week.
Note: There seems to be more than one answer for each problem I and 2, with the assosiciative property of multiplication:
Problem 1:
12 x 6 x 2
(12 x 6) x 2 = 144
(2 x 12 ) x 6 = 144

Problem 2:
11 x 5 x 16
(11 x 5) x 16 = 880
(11 x 16) x 5 = 880

Is this right?

PS: I also wanted to apologize for not posting this in the right section, I tried my best.

 

[Ouabache]

Note: There seems to be more than one answer for each problem I and 2, with the associative property of multiplication:
Problem 1:
12 x 6 x 2
(12 x 6) x 2 = 144
(2 x 12 ) x 6 = 144

Problem 2:
11 x 5 x 16
(11 x 5) x 16 = 880
(11 x 16) x 5 = 880

Is this right?

You are correct in describing the associative property (for multiplication)
and also its application to questions (1) and (2)
and there are more than two answers.

In addition to those you've found, you could have
For (1)
12 x (6 x 2) = 12 x 12 = 144
For (2)
11 x (5 x 16) = 11 x 80 = 880

I'll give you a hint on (3); what you want to determine is your "pay". You might assign that a variable name.
Now, in words, what do you need to do, to calculate your pay? (your description should not include any numbers or variable names)

 

[BadFish]

Sorry, I had to look up the answer. I just couldn't think of an answer.

Let p = pay, then p = 5 x h

Here's another one:
21. Calculate your pay if you work more than 40 hours and are paid time and one half for every hour past 40.

 

[Ouabache]

Sorry, I had to look up the answer. I just couldn't think of an answer.

Let p = pay, then p = 5 x h

Looking up the answer, does not help you learn how to solve the problem.
I was teaching you a strategy for solving these kinds of questions. Before moving on, let's try and understand what happened in part(3)#20. Once you understand the strategy I am sharing, you will be better able to field questions like #21, on your own.

What were the other pieces of information given to calculate the "pay"?

Since you got stuck there, I'll give you a couple more hints.
You were given a variable of "time" (hours) and a "rate" (in units per time) In this case your rate was in dollars per hour. They could have given you a rate in British Pounds per year.

They also ask you to assign a variable to these terms. Often the first letter is a good choice (unless there are more than one term with the same first letter). Let's call "pay" P, and "rate" r, and "time" t.

You now have all the pieces you need to calculate pay. Can you tell me in words, how you to determine "pay"?

 

[BadFish]

pay = rate * time

pay = rt

Is this right?

 

[Ouabache]

Good Job!

Okay in #21, let's approach in a similar fashion.
What is the result you wish to find? (Assign that a variable name).
What other pieces of information do they give you?

 

[BadFish]

21. Calculate your pay if you work more than 40 hours and are paid time and one half for every hour past 40.

p = pay

t = time

t * 1/2

 

[Mark44]

t * 1/2 represents only half the time. Try it with some specific numbers first to get a feel for the kind of calculation you need to do.

If you are paid $10/hr and you work 50 hours in a week, what would be your gross pay?

 

[BadFish]

$10/per hour * 50 = $500 gross pay

 

[symbolipoint]

BadFish, tell us more about the book you are using, Algebra the Easy Way. How does it compare or constrast to a normal introductory textbook on Algebra 1? You could study from a good traditional book on your own over maybe 3 to 5 months. You might then, if you believe you learned well enough, enroll in an Intermediate Algebra course at a community college (or you could study this too, on your own). You would not really need to buy a new book from a retail bookstore, especially if you have a local source of used books. Even books 20 to 35 years old can be excellent.

As far as the number properties ("field axioms") and factoring either plain numbers or expressions, this is well covered in the first few chapters of traditional beginner Algebra books.

 

[BadFish]

BadFish, tell us more about the book you are using, Algebra the Easy Way. How does it compare or constrast to a normal introductory textbook on Algebra 1? You could study from a good traditional book on your own over maybe 3 to 5 months. You might then, if you believe you learned well enough, enroll in an Intermediate Algebra course at a community college (or you could study this too, on your own). You would not really need to buy a new book from a retail bookstore, especially if you have a local source of used books. Even books 20 to 35 years old can be excellent.

As far as the number properties ("field axioms") and factoring either plain numbers or expressions, this is well covered in the first few chapters of traditional beginner Algebra books.

It is an algebra book that explains in a medievil story how algebra works and why. It has traditional axions and properties that a normal book has, only in a story format.

 

[Ouabache]

$10/per hour * 50 = $500 gross pay

Okay, but you are also given a condition of pay, when your work is > 40hrs.
In this practise example, how many hours did you work over 40hrs?
With that amount hours (over 40hrs) , how much would you earn?

 

[BadFish]

I really just don't understand this, the word problems.

I really have no idea...how to determine over 40 hours. I think I would need to go to a class with a teacher who can help me rather than self learn.

I'm going to go read the book again and try to understand it more. I really want to learn this well as I enjoy math and I don't mind the challenge, not going to give up on this.

 

[Ouabache]

I am sorry, I wasn't clear in my last post.
In this practice example, where your total hours worked is 50.
How many hours did you work over 40hrs?

Do you know what time and a half means? (hint: on google search engine, try some key words like "time and a half" . Be sure to use quotation marks if you are search for a specific phrase)? Once you have found that, please let us know here.

 

[BadFish]

Here's another example of it. I just feel awful because I used to be very good at understanding algebra, in class with a teacher but learning it by myself again is very hard for me.

Since multiplication is just a short cut to addition, I assume the rules are the same for multiplication

Rules for adding odd and even numbers:
If you add together two even numbers, the result is an even number
If you add together two odd numbers, the result is an even number
If you add together one even number and one odd number, the result is an odd number

Here's what's confusing me:
"Next, we developed our first result that we proved, rather than assumed. (a proved result is called a theorem.) We set out to prove the addition properties for odd and even numbers. We realized that any even number could be written in the form 2 x n, where n is some natural number. (I think a natural number times a natural number always equals a natural number. if you divide or subtract a natural number it doesn't)

An odd number can be written in the form 2 x n + 1
We tried adding together one even number called 2 x n and another even number called 2 x m, calling the result sum s

s = (2 x m) + (2 x n)
Using the distributive property,
s= 2 x ( m + n)

From the closure property, m + n must be a natural number, so s can be written in the form s = 2 x (some natural number)
Therefore, s must be even
Next, we tried adding together two odd numbers, called 2 x m + 1 and 2 x n + 1 (again calling the result s)
s= (2 x m + 1) + (2 x n + 1)
s = 2 x m + 2 x n + 2
s = 2 x (m + n + 1)

Since m + n + 1 must be a natural number, it follows that s must be even. We had one more combination to do: the sum of an odd number (which we called 2 x m +1 ) and an even number (which we called 2 x n):
s = (2 x n ) + (2 x m + 1 )
s = 2 x (m + n) + 1

Since 2 x (m + n ) must be even, it follows that 2 x (m + n) + 1 must be odd.
"We did it!" The professor exclaimed in amazement. "We can prove general behavior rules by using symbols to stand for letters! I wasn't even sure that could be done!"

---------------

That's an intercept from the book. I wish this would make more sense to me...like it used to.

 

[BadFish]

I am sorry, I wasn't clear in my last post.
In this practice example, where your total hours worked is 50.
How many hours did you work over 40hrs?

Do you know what time and a half means? (hint: on google search engine, try some key words like "time and a half" . Be sure to use quotation marks if you are search for a specific phrase)? Once you have found that, please let us know here.

Hi Quabache, thank you for your help and patience! It surely is appreciated. A few thing about myself. I have always had learning problems...I have a form of autism called aspergers. I remember in pre-algebra having a problem with a square or rectangle, there was an unknown space. You would simply subtract one side from another to get the unknown space. I did not understand this for a long time and it drove me crazy, once I found out how simple it was I understood it and once I "get it" I got it down. Equations are just trying to make each side balanced, I think. By removing something from one side of the equation, you must add it to the other side to make it balanced, which cancels it out. (for example, +5 - 5 = 0)

The first question is easy, 50-40, 10 hours over 40

Hmm, time and a half. Isn't that halve of the total time? i.e let's say total time is 10 minutes. a = 10. Halve would be 10*1/2 = 5 or 10/2 = 5 or a * 1/2 = 5 and so on.

 

[Ouabache]

, thank you for your help and patience! It surely is appreciated.

I'm pleased to help out.

Here's what's confusing me:
"Next, we developed our first result that we proved, rather than assumed. (a proved result is called a theorem.) We set out to prove the addition properties for odd and even numbers.

Very interesting.. Don't worry right now if you don't understand all of this. It will make more and more sense as you go on. Looking through that discussion, I am seeing things that I learned to apply in discrete mathematics in college. I am sure it was right there in my middle school algebra text, it just was never emphasized in my course.

The first question is easy, 50-40, 10 hours over 40

Great job! We will need that number to solve this practice example that Mark44 gave you.

Hmm, time and a half. Isn't that halve of the total time? i.e let's say total time is 10 minutes. a = 10. Halve would be 10*1/2 = 5 or 10/2 = 5 or a * 1/2 = 5 and so on.

That's not quite right. But you did try and explain what you think it means.
So the next step is to find out what "time and a half" does mean.

Are you familiar with using search engines on the net? They are free services where you can put in some information you wish to search on, and it will return possible sites with applicable information. One that I have used is called http://www.google.com" .. There is an open box on their page where you can enter information. In this case we want to choose some key words that can assist in learning what time and a half means. I suggest typing "time and a half" with quotations marks, and then either select the search button with your cursor or just depress the "enter" key on your keyboard. It should return some useful sites. Read through the first few and see how they define "time and a half".
(Why am i spending time teaching how to use search engines?
It is probably the most useful tool that I use on the net. It is a good idea to learn how to use it properly. It can save you lots of time, when learning concepts like this).

 

[BadFish]

Time-and-a-half is when a worker (or workers) is paid 1.5 times their usual hourly rate. It is usually paid as an incentive to work on a particular day (e.g. on Sundays) or as government-mandated compensation for having workers work on particular days (e.g. public holidays).

Is this right?

 

[Mark44]

Yes, this is right. The time-and-a-half rate (1.5 times normal rate) comes in on hours worked over the normal 40 hours per week.

To continue, again if someone gets $10/hour and works 50 hours,

How many hours does he/she get the normal pay rate?
How many hours does he/she get the overtime pay rate?
What's the amount he/she gets at the normal rate?
What's the amount of overtime pay?
What's the total?

 

[Ouabache]

Yes, this is right..

I started to respond to Badfish's last, but you beat me to it.
Since you gave this practice example, it is only fitting you continue.
(hint to Badfish, you calculated the answer to Mark44's 2nd question
"How many hours at the overtime pay rate?" in your post #19).

 

[BadFish]

Yes, this is right. The time-and-a-half rate (1.5 times normal rate) comes in on hours worked over the normal 40 hours per week.

To continue, again if someone gets $10/hour and works 50 hours,

How many hours does he/she get the normal pay rate?
How many hours does he/she get the overtime pay rate?
What's the amount he/she gets at the normal rate?
What's the amount of overtime pay?
What's the total?

How many hours does he/she get the normal pay rate?
A. 40 hours

How many hours does he/she get the overtime pay rate?
B. 10 hours

What's the amount he/she gets at the normal rate?
C. $10/hour * 40 = $400

What's the amount of overtime pay?
D. 10 hours total, so 10 * 10 = 100 dollars

What's the total?
E. $500

 

[Mark44]

A, B, and C are right, but D and E aren't. For the overtime hours, the person gets 1.5 times the normal pay, which is how much per hour?

 

[Mark44]

Where I'm going with this is to step you through the thinking for this kind of problem. If you had to calculate the pay for someone, the information you would need would be the number of hours worked in a week, and the hourly pay rate.

Your formula would have to have two branches: one for anyone working up to 40 hours in the week, and another for anyone who had worked more than 40 hours. Anyone in the latter category would have worked 40 hours at p dollars per hour + some number of hours at 1.5p dollars per hour. Another way to look at the overtime pay is 1.5 times the number of hours times the normal pay per hour. Both would give you the same value.

 

[BadFish]

$15 per hour

Let p = pay per hour then,

p * 40 = (total amount earned)

Overtime
p * 40 + 1.5p = (total amount earned)
How would we write this formula? And how can I understand the other thing I posted better, proving the theorum?

Let me understand this thing first...an odd number is a number where,

a = even number , then

a + 1 = odd number

This is true because when you add an even number and an odd number, you get an odd number?

 

[symbolipoint]

BadFish, in #27, you missed a factor in one of the terms.

Example is: 50 hours of work based on regular rate of 10 dollars per hour; time-and-a-half for anything over 40 hours.

Do you see a sum indicated in the wording? The time is 40 hours at regular pay PLUS 10 hours at the increased pay.
If p=regular pay rate, then gross income is:
40*p + 10*1.5*p

I did not include the units in that expression but you can supply them and analyze the result. In that expression, p = 10. Do not become confused; the regular pay rate given is 10 dollars per hour, and the amount of overtime is 10 hours.

 

[Mark44]

Let me understand this thing first...an odd number is a number where,
a = even number , then
a + 1 = odd number
This is true because when you add an even number and an odd number, you get an odd number?

Yes. If a is an even number, then a + 1 will be an odd number. And if you add an even number and an odd number, you get an odd number. By "number" here, I mean integer, a number with no fractional or decimal part.

 

[Ouabache]

So the point of this digression is to steer you in the direction for developing an equation(s) for the question:

Calculate your pay if you work more than 40 hours and are paid time and one half for every hour past 40.

(note in this original question, you are not given how many hours past 40, a person works, or the normal rate of pay).
Before developing a general expression(s) for this question. Let's see how you making out with Mark44's example.

Discussing the theorem (from your post #18) is another topic. It would be well worthwhile to begin that question on a different thread. (On this forum we discuss one topic at a time per thread).

 

[BadFish]

To continue, again if someone gets $10/hour and works 50 hours,

How many hours does he/she get the normal pay rate?
How many hours does he/she get the overtime pay rate?
What's the amount he/she gets at the normal rate?
What's the amount of overtime pay?
10 hours over over time...
First we need to find out the normal rate.
$10/per hour (h)

1.5 x * 10 = $15, which is our new rate
$15 * 10 hours = $150
What's the total?
$400 + overtime pay (150) = $550

 

[Mark44]

That's right, $550. Now to bring this into the realm of algebra, if someone worked h hours in a week, what expression or expressions represent the total pay?

One quibble with your work in the previous post:

1.5 x * 10 = $15, which is our new rate

Use either x OR * to represent multiplication, but not both. I'm pretty sure you meant that 'x' as "times." Writing 'x' and '*' might be cause the 'x' to be misconstrued as a variable.

 

[BadFish]

:sigh:

I can't think of the answer right now. I'm going to try and figure it out later..maybe when my mind is clear. It's probably so simple, ugh, this is driving me nuts! Not going to give up on it though...

I just wanted to also say a huge thank you for all of you who have had patience with me...it takes me a while to learn things. Once I get it down though, It's good.

 

[Ouabache]

You're welcome.. You're closer than you realize towards finishing this question.
Just take a deep breath (oxygenate to the brain ), collect your thoughts, jot down some ideas and I know you'll have it.

 

[Mark44]

Think about what you would do to calculate total pay for the week for a) 30 hrs, b) 40 hrs, c) 50 hrs. For the last one you have to do something different from what you would do for the first two.