it says 0.667 is closest to 2/3: (why/how?)
0.667 = 66.67/100 (how is it 66.67 where did that number come from, wouldnt it be 667/1000? )
If we simplify by dividing the numerator and denominator by 33.33 we will arrive at 2/3. (help)
6 Answers

The fraction 1/3 is equal to decimal fraction 0.33333_3 where the _3 indicates that 3 repeats forever.
Similarly, 2/3 is twice that, or 0.66666_6 (using the same notation). It is common to round the last digit of interest, so this might be expressed as 0.67, 0.667, 0.6667, and so on, depending on the number of significant digits one wants to use.
Thus, when asked to convert 0.667 to a common fraction, one has to guess whether this is the full precision value of the number (= 667/1000), or whether it is an approximation of the fraction 2/3.
Your text is apparently being a little sloppy when it says that 0.667 = 66.67/100. We know that in order to consider this to be true, we must interpret each side of the equal sign as a representation of 2/3. (Usually, we want to be more rigorous than that in our interpretation of what “=” means.)
_____
Suppose you have a repeating decimal that has a 6digit repeat. (For example, 1/7 = 0.142857_142857. Call this value “n”. Now you can multiply n by 10^6, where the exponent 6 is the number of repeating digits. Then you have
.. n = 0.142857142857_142857
.. 10^6*n = 142857.142857142857_142857
Subtracting n from 10^6*n, we get
… 999999n = 142857 … exactly. The repeating decimal fraction cancels in the subtraction. (You will note there are 6 nines, the number of repeating digits.)
Now, you can find n as a common fraction.
.. n = 142857/999999
Reducing this, we have
.. n = 1/7
This method allows you to turn any repeating decimal into a common fraction.
The _dddd notation used here takes the place of an overbar notation that is more conventional. In a text, 1/3 might be represented as 0.3 where the 3 has an overbar. (There is an error in the example at the second source link. The correct answer is A: 4.1_3 as the working out shows.)

2/3 = 0.6666666…. where the 6’s continue forever
0.667 = 667/1000 which cannot be further reduced.
Some common fractions and there decimal equivalents:
1/2 = 0.5
1/4 = 0.25
1/8 = 0.125
3/4 = 0.75
3/8 = 0.375
5/8 = 0.625
7/8 = 0.875
1/3 = 0.33333…. where the 3’s repeat forever
2/3 = 0.66666…. where the 6’s repeat forever
1/5 = 0.2
2/5 = 0.4
3/5 = 0.6
4/5 = 0.8

1/3

We got that number, .667 by diving 2/3. Though the answer is a repeating .6666666… we round it to .667. Divide it out and you’ll see.
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667/1000.

Yes, 0.667 = 667/1000, but note that
667/1000 = 66.7/100 = 6.67/10 = 66.7/1, etc.