Exercise 58 Page 697

We want to write the repeating decimal 0.7 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	0.7
Sum of Decimals	0.7+0.07+0.007+0.0007+...
Sum of Fractions	7/10+ 7/100+7/1000+7/10 000+...

Consider the sum of fractions above. Note that we can think of it as a geometric series that has a first term of a_1= 710. 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.

a_2÷ a_1

SubstituteII

a_1= 7/10, a_2= 7/100

7/100 ÷ 7/10

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Evaluate

DivFracByFracDiv

a/b÷c/d=a/b*d/c

7/100 * 10/7

SplitIntoFactors

Split into factors

7/10(10) * 10/7

CancelCommonFac

Cancel out common factors

7/10(10) * 10/7

SimpQuotProd

Simplify quotient and product

1/10

Each term of the sequence can be obtained by multiplying the previous term by the common ratio 110.

Let's substitute r= 110 and a_1= 710 in the formula for the sum of an infinite geometric series.

S=a_1/1-r

SubstituteII

r= 1/10, a_1= 7/10

S=710/1- 110

▼

Simplify right-hand side

OneToFrac

Rewrite 1 as 10/10

S=710/1010- 110

SubFrac

Subtract fractions

S=710/910

FracToDiv

a/b=a÷ b

S=7/10 ÷ 9/10

DivFracByFracDiv

a/b÷c/d=a/b*d/c

S=7/10 (10/9)

MultFrac

Multiply fractions

S=70/90

ReduceFrac

a/b=.a /10./.b /10.

S=7/9

We found that the series formed by the sum of the terms is 79. With this information we can express the given number as a single fraction. 0.7=7/9