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.

Number	0.321
Sum of Decimals	0.3+0.021+0.00021+0.0000021+...
Sum of Fractions	3/10+ 21/1000+21/100 000+21/10 000 000+...

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= 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.

S=a_1/1-r

SubstituteII

r= 1/100, a_1= 21/1000

S=211000/1- 1100

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Simplify right-hand side

OneToFrac

Rewrite 1 as 100/100

S=211000/100100- 1100

SubFrac

Subtract fractions

S=211000/99100

FracToDiv

a/b=a÷ b

S=21/1000 ÷ 99/100

DivFracByFracDiv

a/b÷c/d=a/b*d/c

S=21/1000 (100/99)

MultFrac

Multiply fractions

S=2100/99 000

ReduceFrac

a/b=.a /300./.b /300.

S=7/330

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.

0.321=3/10+7/330

ExpandFrac

a/b=a * 33/b * 33

0.321=99/330+7/330

AddFrac

Add fractions

0.321=106/330

ReduceFrac

a/b=.a /2./.b /2.

0.321=53/165