# How do you convert 0.83 (3 repeating) to a fraction?

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To format this question type you write: 0.8bar3

But you use the hash key just before 0.8bar3

and also at the end. So you end up with

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let

Then

So

Multiply both sides by 10

Divide both sides by 90

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Here's a method using a calculator to help...

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Here's another way you can convert decimals to fractions if you have a calculator to hand.

We use the calculator to find the terminating continued fraction expansion for the given number, then unwrap it to a regular fraction.

For our example, type

Note that the portion before the decimal point is

#color(blue)(0) + #

Take the reciprocal of the given number to get a result something like:

#color(blue)(0) + 1/color(blue)(1)#

then subtract it to get

#color(blue)(0) + 1/(color(blue)(1)+1/color(blue)(5)) = 0+1/(6/5) = 5/6#

**Another example**

Just to make the method a little clearer, let us consider a more complex example:

Given:

#3.82857142857#

Note the

#1.20689655173#

Note the

#4.83333333320" "color(lightgrey)"Note the rounding error"#

Note the

#1.20000000019#

Note the

#4.99999999525#

Let's call that

Taking the numbers we have found, we have:

#3.82857142857 = color(blue)(3) + 1/(color(blue)(1)+1/(color(blue)(4)+1/(color(blue)(1)+1/color(blue)(5))))#

#color(white)(3.82857142857) = color(blue)(3) + 1/(color(blue)(1)+1/(color(blue)(4)+5/6))#

#color(white)(3.82857142857) = color(blue)(3) + 1/(color(blue)(1)+6/29)#

#color(white)(3.82857142857) = color(blue)(3) + 29/35#

#color(white)(3.82857142857) = 134/35#

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