who is right here? 1 + 1/(1 + 1/2) = 5/3 (but I get 4/5)

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allegansveritatem

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Here is one of the choices in a multiple choice problem whose question is: Which of the following is true? This is choice C:

. . . . .\(\displaystyle \mbox{C. }\, 1\, +\, \dfrac{1}{1\, +\, \frac{1}{2}}\, =\, \dfrac{5}{3}\)

I tried all the other choices and they all seemed to be false. But I don't seem to be able to come up with (5)/(3) when I do the preoblem. Here is what I did:

solution.jpg

I spent quite a lot of time on this and did it several ways and in one of the attempts I did come up with (5)/(3), but I finally settled on the answer that is in the image above. Am I wrong or is one of the other choices right and I am wrong in rejecting them? I will see if I can find a photo of the whole problem and I will upload it below if I can find it.

I found it! Here is the whole thing:

whole problem.jpg
 

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allegansveritatem said:
Here is one of the choices in a multiple choice problem whose question is: Which of the following is true? This is choice C:

View attachment 10425

I tried all the other choices and they all seemed to be false. But I don't seem to be able to come up with (5)/(3) when I do the preoblem. Here is what I did:

View attachment 10426

I spent quite a lot of time on this and did it several ways and in one of the attempts I did come up with (5)/(3), but I finally settled on the answer that is in the image above. Am I wrong or is one of the other choices right and I am wrong in rejecting them? I will see if I can find a photo of the whole problem and I will upload it below if I can find it.

I found it! Here is the whole thing:

View attachment 10427
Click to expand...

That last picture tells me that you have been taught two ways to simplify an expression like this. Did you try doing what you were taught?

Your error is on the second line of your work. When you add two fractions, you can't just add the numerators and add the denominators. That is how we multiply fractions; but you can only add fractions when they have the same denominator, and then we add the numerators and keep the same denominator.
 
Make sure when you write these down that you make enough room to keep the fractions in the fractions easy to read. This kind of problem can cause a lot of confusion otherwise.

\(\displaystyle \dfrac{1}{1 + \dfrac{1}{2}} = \dfrac{1}{1 + \dfrac{1}{2}} \cdot \dfrac{2}{2}\)

\(\displaystyle = \dfrac{1 \cdot 2}{ 2 \left ( 1 + \dfrac{1}{2} \right ) }\)

\(\displaystyle = \dfrac{2}{2 \cdot 1 + 2 \cdot \dfrac{1}{2} }\)

\(\displaystyle = \dfrac{2}{2 + 1} = \dfrac{2}{3}\)

-Dan
 
These last two posts demonstrate the two methods you will have been taught:
  • rewriting the denominator as a single fraction and multiplying by its reciprocal, and
  • multiplying the numerator and denominator by the LCD (in this case, 2).

Be sure to learn these methods well, so that you can use them when needed, and not fall back on invalid "methods".
 
allegansveritatem said:
Here is one of the choices in a multiple choice problem whose question is: Which of the following is true? This is choice C:

. . . . .\(\displaystyle \mbox{C. }\, 1\, +\, \dfrac{1}{1\, +\, \frac{1}{2}}\, =\, \dfrac{5}{3}\)

I tried all the other choices and they all seemed to be false.
Click to expand...
Don't try to do the whole thing at once. Start inside, and work your way out:

. . . . .\(\displaystyle 1\, +\, \frac{1}{2}\, =\, \frac{2}{2}\, +\, \frac{1}{2}\, =\, \frac{3}{2}\)

This is under just a "1", so the fraction in the bottom (the 3/2) flips upside down and multiplies itself against the 1, which obviously leads to:

. . . . .\(\displaystyle \dfrac{1}{\left(\frac{3}{2}\right)}\, =\, 1\, \times\, \frac{2}{3}\, =\, \dfrac{2}{3}\)

Then add in the remaining "1":

. . . . .\(\displaystyle 1\, +\, \dfrac{2}{3}\, =\, \dfrac{3}{3}\, +\, \dfrac{2}{3}\, =\, \dfrac{5}{3}\)

I don't know where the "x^2" bits come from, in your third line of "work" displayed. But the very first step was wrong, because it is not possible to add fractions by adding their denominators. For instance, 1/2 plus 1/2 ought to be 1 whole. But, using your method, you would instead, get (1+1)/(2+2) = 2/4 = 1/2, which is wrong. ;)
 
Dr.Peterson said:
That last picture tells me that you have been taught two ways to simplify an expression like this. Did you try doing what you were taught?

Your error is on the second line of your work. When you add two fractions, you can't just add the numerators and add the denominators. That is how we multiply fractions; but you can only add fractions when they have the same denominator, and then we add the numerators and keep the same denominator.
Click to expand...

Oh Boy! I don't know how I could have done that. And done it. And done it. I know these things! But.... I haven't got a leg to stand on. Thanks for pointing this out.
 
topsquark said:
Make sure when you write these down that you make enough room to keep the fractions in the fractions easy to read. This kind of problem can cause a lot of confusion otherwise.

\(\displaystyle \dfrac{1}{1 + \dfrac{1}{2}} = \dfrac{1}{1 + \dfrac{1}{2}} \cdot \dfrac{2}{2}\)

\(\displaystyle = \dfrac{1 \cdot 2}{ 2 \left ( 1 + \dfrac{1}{2} \right ) }\)

\(\displaystyle = \dfrac{2}{2 \cdot 1 + 2 \cdot \dfrac{1}{2} }\)

\(\displaystyle = \dfrac{2}{2 + 1} = \dfrac{2}{3}\)

-Dan
Click to expand...

I admit I need to improve my penmanship in this area.
 
stapel said:
Don't try to do the whole thing at once. Start inside, and work your way out:

. . . . .\(\displaystyle 1\, +\, \frac{1}{2}\, =\, \frac{2}{2}\, +\, \frac{1}{2}\, =\, \frac{3}{2}\)

This is under just a "1", so the fraction in the bottom (the 3/2) flips upside down and multiplies itself against the 1, which obviously leads to:

. . . . .\(\displaystyle \dfrac{1}{\left(\frac{3}{2}\right)}\, =\, 1\, \times\, \frac{2}{3}\, =\, \dfrac{2}{3}\)

Then add in the remaining "1":

. . . . .\(\displaystyle 1\, +\, \dfrac{2}{3}\, =\, \dfrac{3}{3}\, +\, \dfrac{2}{3}\, =\, \dfrac{5}{3}\)

I don't know where the "x^2" bits come from, in your third line of "work" displayed. But the very first step was wrong, because it is not possible to add fractions by adding their denominators. For instance, 1/2 plus 1/2 ought to be 1 whole. But, using your method, you would instead, get (1+1)/(2+2) = 2/4 = 1/2, which is wrong. ;)
Click to expand...

Yes, I know now that I added the denominators. And I did not do a very good job with recording my working out of the problem. What looks like an exponent is really an LCD.
 
I want to thank all who responded and all who contributed different ways to approach this problem. I am going to try some of them tomorrow. Too late to think straight now.
 
Here is how I went about this problem today:

solution.jpg

I like this method more than the multiply by reciprocal approach. Anyway, finally!
 
Just a quick comment. I don't know how your instructors do it but you apparently have a 1 1/2 in your denominator? (Or is it a typo?) Mixed fractions like this will get you into a lot of trouble. I strongly recommend never using mixed fractions. Always write it out as 1 + 1/2.

-Dan
 
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