Sometimes a calculator is more complicated than it needs to be. For example, the display of

7.33333333333 |

is almost certainly. This is easy to recognise, but what about

0.09090909091 |

It should not take you too long to recognise this as

**Problem 1** Rewrite the decimal numbers below as fractions:

**a)** 0.6666666667 **b)** 0.3636363636
** c)** 0.5555555556 ** d)** 0.1428571429.

The first three are all relatively straightforward, but **d)** is a much
more complicated recurring decimal. Its cycle consists of 6 recurrent
numbers (142857). The number is in fact a decimal approximation to
Check
this on your calculator.

Now consider

0.0958904109589 |

You have to be very familiar with decimal approximations to spot this one!

There is, though, a method for finding the fraction equivalent. We will illustrate it with this example.

Let
*x* = 0.0958904109589...

so that
10^{8
}*x* = 9589041.09589...

Subtracting the first equation from the second gives

**Problem 2** Cancel out common factors in the expression
for *x* to find its fraction value in lowest terms.

**Problem 3** Use the same procedure to find the fraction equivalent
of

**a)** 0.5714285714... **b)** 0.027027027... **
c)** 0.07692307692...

**EXTENSION** Use a computer, a calculator, or long division by hand,
to find the recurring decimals for all fractions

Which fraction has the longest cycle?