Write the given number as a sum of decimals. Then, write those decimals as fractions.
6 727
Practice makes perfect
We want to write the repeating decimal 6.259 as a fraction. To do so, we will start by writing this number as a sum of decimals and as a sum of fractions.
Number
6.259
Sum of Decimals
6+0.259+0.000259+0.000000259+...
Sum of Fractions
6/1+ 259/1000+259/1 000 000+259/1 000 000 000+...
Consider the sum of fractions above. Note that from the 2^(nd) term on,
we can think of it as a geometric series that has a first term of a_1= 2591000. To find its common ratio r, we can divide any term of the sequence by its previous term. For simplicity, we will divide a_2 by a_1.
We found that the series formed by the sum of the terms, starting from the 2^(nd),
is 727. With this information we can express the given number as a sum of two fractions.
6.259=6/1+7/27
Finally, we will add the fractions to obtain the value of 6.259 expressed as a single fraction.
