Exercise 35 Page 687

We want to write the repeating decimal 0.321 as a fraction. To do so, we will start by writing this number as a sum of decimals and as a sum of fractions.

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= 211000. Let's find its common ratio r.

The common ratio is r= 1100. Let's substitute this value — together with a_1= 211000 — in the formula for the sum of an infinite geometric series.

We found that the series formed by the sum of the terms, starting from the 2^(nd), is 7330. With this information we can express the given number as a sum of two fractions. 0.321=3/10+7/330 Finally, we will add the fractions to obtain the value of 0.321 expressed as a single fraction.