Question

Hi I need to write the equations for the following please help:
I also have done the solutions against them, please check them and let me know if they are right:
1. the sum of multiplicative inverses of two positive even integers is 5/12. Find the integers.

2. Find the natural numbers which when added to its square gives 9 times the next natural number.

Answer 1

3. Joe bought some chairs for Rs. 200. 5 chairs were stolen from the total. He sold the remaining chairs for a profit of Rs. 40. The selling price of each chair was Rs. 4 more than its cost price. How many chairs did he buy and how much was the cost price of each chair?

Answer 2

4. The denominator of a fraction is 4 more than its numerator. Taking the square of this fraction we get a new fraction. The difference between the denominator and numerator of the new fraction is 40. Find the fraction.

Answer 3

Here are my answers now:
1. \[(1/x) - (1/x+2) = 5/12\]

Answer 4

what is Rs ?

Answer 5

2. x \[x + x ^{2} = 9 (x+1)\]

Answer 6

1. is wrong
2. is right
...

Answer 7

Rs. (Rupees) is a currency similar to Dollars.

Answer 8

Sorry, should have known.

Answer 9

what is the multiplicitive inverse of a number?

Answer 10

If x is a number, it's multiplicative inverse is 1/x

Answer 11

right, so we have two positive even integers. But it doesn't say "successive" so we can't write them as x and x+2
it also says "sum" not "difference"
can you fix it do you think?

Answer 12

Oh yeah I wrote the minus sign by mistake, but I'm unable to figure out how to express those two even integers?

Answer 13

Start off calling the two numbers 'a' and 'b'.
that gives us\[\frac{1}{a}+\frac{1}{b}=\frac{5}{12}\]but we know that these numbers are even, so we can write them as.
a=2n
b=2m
because any even number is 2 times SOME integer.
So what will you get after using that?

Answer 14

1/2n + (1/2m) = 5/12

But then how would we solve the answer for x?

Answer 15

we don't it would seem
do you know that you need to solve it?

Answer 16

Yes, i have to make the equations, and then solve for x. The answer given at the end of the book for the question is 4 and 6.

Answer 17

I think 2x and 2x+2 as the even integers, even if they donts say successive even numbers, we dont have any other solution for it.

Answer 18

dont*

Answer 19

This is what is called a Diophantine equation.
That means the answers can only be integers and the number of variables is more than the number of solutions.
Since the answers are 4 and 6 that would suggest that they meant to write the word 'successive' and didn't.
Seems to be a mistake in the book.

Answer 20

If they mean successive then your formula is correct (with the + fixed).
Note that if we call them 2x and 2x+2 we need to multiply x by 2 at the end to find the actual numbers.

Answer 21

Yes, definitely seem sto be an error in the book. Ok, now for the third one, i'm pretty clueless.

Answer 22

gotta read it one sec...

Answer 23

Ok first we need a formula for how much he spent.
Do you have that?

Answer 24

He spent $ 200.

Answer 25

He spent $ 200.

Answer 26

he bought x chairs, 5 weres tolen so,
x-5

Answer 27

you know, again the wording is weak
is it 200 total? or 200 per chair?
it's rupees so I thought 200 per chair, right?

Answer 28

He spent $ 200.

Answer 29

No, "Joe bought some chairs for $ 200"

Answer 30

He spent $ 200.

Answer 31

so "some" is definitely "x"

Answer 32

yes you are right, sorry...

Answer 33

He spent $ 200.

Answer 34

no problem, so first expression is going to be "x-5"

Answer 35

right so we need to represent 200 by two variables, the one for price and the one for number of chairs.
can you do that?
200=???

Answer 36

no not x-5 first because we need two variables here to represent the two different things.

Answer 37

x for chairs and y for 200

Answer 38

Not quite, we will represent
200=Px
where P is the price per chair and x is the number of chairs.
Make sense?

Answer 39

ok so, x-5 = Px?

Answer 40

not quite...
remember he sold the number of chars after 5 were stolen for 4 MORE than the original price P.
so what does Px become? (both P and x change a bit)

Answer 41

x-5 = Px +4
But then where did 200 go?

Answer 42

close...
200=Px
selling them we are told he changed the price by +4 and the thief changed the number of chairs by -5. We also know he MADE $40 total. So we have

purchase:
200=Px
selling:
200+40=(P+4)(x-5)

reasonable?

can you solve those two equations?

Answer 43

remember (price per chair)(# of chairs)=(money spent/made)

Answer 44

I cannot solve th eequation actually, i 'm stuck on 4x+px-5p-260=0

Answer 45

Let me try...

Answer 46

The answer at the end of the book is 25 and 8.

Answer 47

he bought 25 chairs
And the Cost of each chair was $8

Answer 48

yes that is the solution that makes sense, now to prove it to you...
it'll take a little work

Answer 49

sales:
200+40=(p+4)(x-5)

remember that 200=Px:

Px+40=Px+4x-5P-20
60=4x-5P

and we can write P=200/x
subbing this in gives

60=4x-5(200/x)=4x-1000/x
60x=4x^2-1000
0=4x^2-60x-1000

can you solve it now?
you will get two solutions, one will be negative and therefor make no sense. Throw that one away and keep the non-negative one.

Answer 50

I think I made an error...

Answer 51

yes, sorry, what did I do?
the equations are correct originally I checked...

Answer 52

Yes, I can!! Thank you so much TT for helping and explaining each and everything so elaborately, and for taking out time for me. I hope i can give you a medal for each of your reply. and each of the secon you spend at the post.

Answer 53

Ok
I wish I knew what I did wrong
If yo put the equations
240=(p+4)(x-5)
and
200=Px into wolfram you can see you get the right answer.

http://www.wolframalpha.com/input/?i=200%3Dpx%26%26240%3D%28p%2B4%29%28x-5%29

So I guess good luck figuring out my mistake

As for the last problem I don't think I can solve it...
Hard!

Answer 54

the fraction is 3/7