Converting decimals to fractions

[Cliff Hanley]

I came across the following question on the BBC website;

Convert 0.272727 to a fraction. It was a multiple choice question so I could test each possible answer by dividing the numerator by the denominator until I got the right one. But it made me wonder how I could have answered it had it not been multiple choice. I could have got as far as 272,727/1,000,000 but then how would I know how to reduce that? I know to look for common factors but would struggle with such a large number.

There was another example also; convert 0.6121212 to a fraction.

 

[mfb]

0.272727 = 27/100 + 27/10000 + 27/1000000 = 27*( ... )

 

[PeroK]

I came across the following question on the BBC website;

Convert 0.272727 to a fraction. It was a multiple choice question so I could test each possible answer by dividing the numerator by the denominator until I got the right one. But it made me wonder how I could have answered it had it not been multiple choice. I could have got as far as 272,727/1,000,000 but then how would I know how to reduce that? I know to look for common factors but would struggle with such a large number.

There was another example also; convert 0.6121212 to a fraction.

Suppose you take a two-digit number: 27 for example. Let's look at 2700/99. For every 100, 99 goes into it once and leaves a remainder of 1. So, when you divide 2700 by 99, you get 27 plus a remainder of 27:

2700 = (27 × 99) + 27

Therefore:

2700/99 = 27.272727...

and:

27/99 = 0.272727...

It's the same for any two digit number:

35/99 = 0.353535...

06/99 = 0.060606...

For 1-digit numbers you have:
1/9 = 0.11111...
2/9 = 0.22222...
3/9 = 0.33333...

Etc.

And for 3-digit numbers you have:

001/999 = 0.001001001...

123/999 = 0.123123123

etc.

 

[Mark44]

Was the problem to convert .272727 to a fraction or was it written as .272727...; i.e., with the dots, indicating that the same pattern repeats infinitely?

 

[HallsofIvy]

Equivalently, if we let x= 0.27272727... then 100x= 27.272727... Now subtracting 100x- x leaves the integer 27 (the fact that the "27" part of the decimal is infinitely repeating means that we will always have the "27"s continuing after the decimal point so they all cancel). That is, 100x- x= 99x= 27 so x= 27/99.

Every fraction is equivalent to a repeating decimal.

(We count 1/2= 0.5 as 0.500000... with the "0" repeating. Every fraction with denominator having only "2"s and "5"s as factors of the denominator is such a "terminating" decimal.)

 

[Mark44]

Convert 0.272727 to a fraction.

Equivalently, if we let x= 0.27272727...

These are two different and unrelated problems. Possibly the OP is unaware of the difference between .272727 and .272727...

We should hold off on further responses until the OP returns to clarify what he is asking.

 

[HallsofIvy]

Thank you. I completely missed that!

 

[HallsofIvy]

I came across the following question on the BBC website;

Convert 0.272727 to a fraction. It was a multiple choice question so I could test each possible answer by dividing the numerator by the denominator until I got the right one. But it made me wonder how I could have answered it had it not been multiple choice. I could have got as far as 272,727/1,000,000 but then how would I know how to reduce that? I know to look for common factors but would struggle with such a large number.

There was another example also; convert 0.6121212 to a fraction.

Surely you know that 1,000,000 is a power of 10, $10^6$? And that 10= 2*5? So the denominator has factors of only 2 and 5. Do 2 or 5 divide 272727?