# Thread: [SOLVED] Calling Mr Fantastic,help and tie-breaker needed

1. ## [SOLVED] Calling Mr Fantastic,help and tie-breaker needed

Mr Fantastic, I really appreciated your clear explanation with my other questions, and hope that you will help me with these too!

I need to check my answers to these two questions:

simplify by dividing:

__________________________________________________ _
I'm having a bit trouble with this question:

the sum of a number and its reciprocal is 10/3. what is the number?
__________________________________________________ _
I also need to make sure that my method is correct and the answer is sound, for this question.

One eighth of the square root of seven less than a number is two. What is the number?
My answer is 263. I came to this answer by working it backward, if 2*8=16, then 16*16=256 and 256+7=263. So if the equation version of the word problem was: (1/8)sqrtN-7=2 then the answer is: (2*8)^2+7=263

Finally, a little while back I posted the following question:

Originally Posted by haflore
I have a very strange proportion that I need to solve.

the question is as follows:

Complete. If x/y=3/5, then x/3=_______

I have deduced that the answer must be either,3/3 or y/5, am I correct?

Thanks in advance for the help!
To which I got these two replies:

Originally Posted by galactus
It would appear that x=3. Then, x/3=1
Originally Posted by icemanfan
The correct answer is y/5. In order to arrive at that conclusion, multiply both sides of the original equation by y/3.

2. I may not be Mr. Fantastic, but I will help you with at least one of these questions...

the sum of a number and its reciprocal is 10/3. what is the number?
The reciprocal of a number is one over that number, (x -> 1/x), so:

$x+ \frac{1}{x} = \frac{10}{3}$

Multiply the entire equation by 3x, the LCD

$3x^2 + 3 = 10x$

$3x^2-10x+3=0$

$x = \frac{10 \pm \sqrt{100 - 4(3)(3)}}{6} \Rightarrow \frac{10 \pm \sqrt{64}}{6} = \frac{10 \pm 8}{6} \Rightarrow \frac{5 \pm 4}{3} = \frac{1}{3}, 3$

3. Hello,

I'm nor Mr, nor Fantastic, nor Mr Fantastic... But I'll try to help ya, k ?

Originally Posted by haflore
Mr Fantastic, I really appreciated your clear explanation with my other questions, and hope that you will help me with these too!

I need to check my answers to these two questions:

simplify by dividing:

Ok

A little typo or mistake, it's not -9 but +9

__________________________________________________ _
I'm having a bit trouble with this question:

the sum of a number and its reciprocal is 10/3. what is the number?
Let x be this number and $\frac 1x$ its reciprocal.

We have $x+\frac 1x=\frac{10}{3}$

Multiplying by x, and assuming it's $\neq 0$:
$\implies x^2+1=\frac{10}{3} \cdot x \implies x^2-\frac{10}{3} x+1=0$

Can you go through it ?

__________________________________________________ _
I also need to make sure that my method is correct and the answer is sound, for this question.

One eighth of the square root of seven less than a number is two. What is the number?
My answer is 263. I came to this answer by working it backward, if 2*8=16, then 16*16=256 and 256+7=263. So if the equation version of the word problem was: (1/8)sqrtN-7=2 then the answer is: (2*8)^2+7=263
I don't like much doing it backward, but I don't know how you've learnt it. Your result is correct, but I'd do this way :

Taking the problem from the end ~
"seven less than a number" : this means N-7
"square root of ..." : this means sqrt(N-7)
"one eighth of the ..." : this means (1/8)sqrt(N-7)
And this is equal to 2, which yields your equation

Finally, a little while back I posted the following question:

To which I got these two replies:
I agree with icemanfan...

Complete. If x/y=3/5, then x/3=_______
Multiplying by y :
x=3y/5

Making appear x/3, by dividing each side by 3 :
x/3=(3y)/(5*3)=y/5

4. Wow! Those were some fast replies! Thanks very much Moo and colby2152,

5. Originally Posted by haflore

I need to check my answers to these two questions:

simplify by dividing:

not sure I agree.

Synthetic Division

Following the math, I get no remainder:

$x^2 + 3x + 6$

if your constant term was positive 6, I'd get a remainder of 12/(x-1)

Solving the cubic the long way here, I get 1 as the real root, which means there should be no remainder when you divide by x minus root:

Soving Cubic Equations

6. Originally Posted by mathceleb
not sure I agree.

Synthetic Division

Following the math, I get no remainder:

$x^2 + 3x + 6$

[snip]
Agreed.

And thanks to all who filled my somewhat small shoes ....... Only problem is that those shoes have been stretched and are now too big for me .....

7. Originally Posted by colby2152
I may not be Mr. Fantastic, but I will help you with at least one of these questions...

The reciprocal of a number is one over that number, (x -> 1/x), so:

$x+ \frac{1}{x} = \frac{10}{3}$

Multiply the entire equation by 3x, the LCD

$3x^2 + 3 = 10x$

$3x^2-10x+3=0$

$x = \frac{10 \pm \sqrt{100 - 4(3)(3)}}{6} \Rightarrow \frac{10 \pm \sqrt{64}}{6} = \frac{10 \pm 8}{6} \Rightarrow \frac{5 \pm 4}{3} = \frac{1}{3}, 3$

Thank you for your invaluable help, I understand everything except; why 3x is the LCD, I can understand either 3 or x being the LCD but 3x seems to suggest that 10/3 was multiplied by 1/x ???.

8. Originally Posted by haflore
Thank you for your invaluable help, I understand everything except; why 3x is the LCD, I can understand either 3 or x being the LCD but 3x seems to suggest that 10/3 was multiplied by 1/x ???.
$\frac{x}{1} + \frac{1}{x} = \frac{10}{3}$

The denominators are: $1, x, 3$

The LCD is the product of these three: $3*1*x = 3x$

This way, when you multiply out by the LCD, all fractions will disappear and the polynomial will simpler to solve via quadratic equation.