Could someone please explain how you could get 6.6667 as 6 2/3?

Could someone please explain how you could get 6.6667 as 6 2/3?

It is not exactly equal, so you would have to use a method that finds a simple fraction that is approximately equal to the decimal. (It is exactly equal to 6 6667/10000.)

The easiest method is "by inspection": You just recognize that 6.6667 is a rounded form of 6.666..., and either from memory or by the standard method, convert that repeating decimal to a fraction.

If you are looking for an algorithm that would allow a computer, with no human insight, to do it, you can find one here: http://mathforum.org/dr.math/faq/faq.fractions.html#decfrac (see section IIC, "Fractions with Small Denominators"). Excel presumably does this when you format a cell as a Fraction; you specify how large the denominator is allowed to be, which determines how close the displayed fraction is.
 
Thankyou. This was an answer to a problem, and I could not figure out how itwas converted from 6.6667 to 6 2/3. Myanswer was 6 33/50. I did not know ifthere was a way by hand to quickly convert the answer as 6 2/3, because obviouslyI could not do it. This leads to myconfusion of how to reach this type of answer by hand or quickly with astandard calculator.
 
Thankyou. This was an answer to a problem, and I could not figure out how it was converted from 6.6667 to 6 2/3. My answer was 6 33/50. I did not know if there was a way by hand to quickly convert the answer as 6 2/3, because obviously I could not do it. This leads to my confusion of how to reach this type of answer by hand or quickly with a standard calculator.

In this setting, you should simply know your decimals well enough to recognize it as 2/3.

But very likely, the best way to solve the problem is not to use decimals at all. I see this too often: a student uses a calculator to solve a problem, getting a decimal answer, when a fractional answer is required; then they struggle to convert their decimal to a fraction. What they should have done (assuming the problem explicitly asked for a fraction) is to do all the work in fraction form (many calculators can even do that for you). Then no conversion is ever needed.

If the problem didn't say you had to give a fraction, then your decimal answer would be correct. If you wanted to see whether your answer matches the fractional form in the back of the book, you could just convert their answer to a decimal: 6 2/3 = 6.6666666..., and you can see that your answer is a rounded form of that. So it is correct!

What did the original problem say?
 
Thankyou. This was an answer to a problem, and I could not figure out how itwas converted from 6.6667 to 6 2/3. Myanswer was 6 33/50. I did not know ifthere was a way by hand to quickly convert the answer as 6 2/3, because obviouslyI could not do it. This leads to myconfusion of how to reach this type of answer by hand or quickly with astandard calculator.
The only fractions that can be expressed exactly in decimal notation are those with denominators that are a power of 2 times a power of 5. No other fraction can be expressed exactly in decimal notation.

What this means is that hand calculators will not give you exact answers to a great many problems. This does not lead to practical problems because modern hand calculators give answers to a great many decimal places and so are accurate enough for most practical purposes. However, if you are studying fractions and must give exact answers, you need to avoid putting fractions into the calculator.

Without seeing the problem, we cannot tell you what went wrong. 33/50 is very close to 2/3, but it is not equal.

\(\displaystyle \dfrac{2}{3} - \dfrac{33}{50} = \dfrac{2 * 50}{3 * 50} - \dfrac{33 * 3}{50 * 3} = \)

\(\displaystyle \dfrac{100}{150} - \dfrac{99}{150} = \dfrac{1}{150} > 0.\)
 
Here is the original problem:

If 50 bottles of soda can be made in 4/3 hour, how manyhours will it take to make 250 bottles of soda?
 
Here is the original problem:

If 50 bottles of soda can be made in 4/3 hour, how many hours will it take to make 250 bottles of soda?

Okay, so you probably divided 250 by 50 and saw that it would take 5 times 4/3 hour, then you did the multiplication using your calculator to get \(\displaystyle 5 \times 4 \div 3 = 20 \div 3 = 6.6667\). What you needed to do (if you were told to use fractions, or if that is a class rule) is to multiply as fractions: \(\displaystyle 5 \times \frac{4}{3} = \frac{5 \times 4}{3} = \frac{20}{3} = 6 \frac{2}{3}\).
 
I setup it as a ratio and went from there. Thank you for the explanation, I sincerely appreciate it.
 
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