The repeating decimal, 0.ModifyingAbove 27 with bar, is converted to the fraction 0.ModifyingAbove 27 with bar = StartFraction 3 Over x EndFraction. What is the value of x in the fraction?

9514 1404 393

  x = 11

Step-by-step explanation:

A 2-digit repeating decimal fraction will have two 9s as the denominator. In this case the fraction can be reduced.

Comparing this to 3/x, we see that x = 11.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4