Exercise 59 Page 697

We want to write the repeating decimal 5.126 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	5.126
Sum of Decimals	5+0.126+0.000126+0.000000126+...
Sum of Fractions	5/1+ 126/1000+126/1 000 000+126/1 000 000 000+...

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= 1261000. 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= 126/1000, a_2= 126/1 000 000

126/1 000 000 ÷ 126/1000

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Evaluate

DivFracByFracDiv

a/b÷c/d=a/b*d/c

126/1 000 000 * 1000/126

SplitIntoFactors

Split into factors

126/1000(1000) * 1000/126

CancelCommonFac

Cancel out common factors

126/1000(1000) * 1000/126

SimpQuotProd

Simplify quotient and product

1/1000

Each term of the sequence can be obtained by multiplying the previous term by the common ratio 11000.

Let's substitute r= 11000 and a_1= 1261000 in the formula for the sum of an infinite geometric series.

S=a_1/1-r

SubstituteII

r= 1/1000, a_1= 126/1000

S=1261000/1- 11000

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Simplify right-hand side

OneToFrac

Rewrite 1 as 1000/1000

S=1261000/10001000- 11000

SubFrac

Subtract fractions

S=1261000/9991000

FracToDiv

a/b=a÷ b

S=126/1000÷ 999/1000

DivFracByFracDiv

a/b÷c/d=a/b*d/c

S=126/1000(1000/999)

MultFrac

Multiply fractions

S=126 000/999 000

ReduceFrac

a/b=.a /9000./.b /9000.

S=14/111

We found that the series formed by the sum of the terms, starting from the 2^(nd), is 14111. With this information we can express the given number as a sum of two fractions. 5.126=5/1+14/111 Finally, we will add the fractions to obtain the value of 5.126 expressed as a single fraction.

5.126=5/1+14/111

ExpandFrac

a/b=a * 111/b * 111

5.126=555/111+14/111

AddFrac

Add fractions

5.126=569/111

FracToMixed

Write fraction as a mixed number

5.126=5 14111