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


is almost certainly7 1/3.  This is easy to recognise, but what about


It should not take you too long to recognise this as 1/11

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 1/7Check this on your calculator.

Now consider


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                                    108 x = 9589041.09589...

Subtracting the first equation from the second gives

(10^8 - 1)x = 9589041 => x = 9589041/99999999

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

1/1, 1/2, 1/3, ..., 1/19

Which fraction has the longest cycle?

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