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McGraw Hill Glencoe Algebra 2, 2012

MH

McGraw Hill Glencoe Algebra 2, 2012 View details

4. Infinite Geometric Series

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Exercise 40 Page 687

Write the given number as a sum of decimals. Then, write those decimals as fractions.

$\frac{4 7 7}{1 1 0 0}$

Practice makes perfect

We want to write the repeating decimal $0.43 \overline{36}$ 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	$0.43 \overline{36}$
Sum of Decimals	$0.43 + 0.0036 + 0.000036 + 0.00000036 + \dots$
Sum of Fractions	$\frac{4 3}{1 0 0} + \frac{3 6}{1 0 0 0 0} + \frac{3 6}{1 0 0 0 0 0 0} + \frac{3 6}{1 0 0 0 0 0 0 0 0} + \dots$

Consider the above sum of fractions. 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} = \frac{3 6}{1 0 0 0 0} .

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} \div a_{1}

$a_{2} = \frac{3 6}{1 0 0 0 0 0 0}$ , $a_{1} = \frac{3 6}{1 0 0 0 0}$

\frac{3 6}{1 0 0 0 0 0 0} \div \frac{3 6}{1 0 0 0 0}

▼

Evaluate

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{3 6}{1 0 0 0 0 0 0} \cdot \frac{1 0 0 0 0}{3 6}

SplitIntoFactors

Split into factors

\frac{3 6}{1 0 0 0 0 ( 1 0 0 )} \cdot \frac{1 0 0 0 0}{3 6}

CancelCommonFac

Cancel out common factors

\frac{3 6}{1 0 0 0 0 ( 1 0 0 )} \cdot \frac{1 0 0 0 0}{3 6}

Simplify quotient and product

\frac{1}{1 0 0}

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

\frac{1}{1 0 0} .

Let's substitute

r = \frac{1}{1 0 0}

and

a_{1} = \frac{3 6}{1 0 0 0 0}

in the formula for the sum of an infinite geometric series.

S = \frac{a _{1}}{1 - r}

$r = \frac{1}{1 0 0}$ , $a_{1} = \frac{3 6}{1 0 0 0 0}$

S = \frac{\frac{3 6}{1 0 0 0 0}}{1 - \frac{1}{1 0 0}}

▼

Simplify right-hand side

Rewrite $1$ as $\frac{1 0 0}{1 0 0}$

S = \frac{\frac{3 6}{1 0 0 0 0}}{\frac{1 0 0}{1 0 0} - \frac{1}{1 0 0}}

Subtract fractions

S = \frac{\frac{3 6}{1 0 0 0 0}}{\frac{9 9}{1 0 0}}

$\frac{a}{b} = a \div b$

S = \frac{3 6}{1 0 0 0 0} \div \frac{9 9}{1 0 0}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

S = \frac{3 6}{1 0 0 0 0} (\frac{1 0 0}{9 9})

Multiply fractions

S = \frac{3 6 0 0}{9 9 0 0 0 0}

$\frac{a}{b} = \frac{a / 3 6 0 0}{b / 3 6 0 0}$

S = \frac{1}{2 7 5}

We found that the series formed by the sum of the terms, starting from the

2^{nd},

is

\frac{1}{2 7 5} .

With this information we can express the given number as a sum of two fractions.

0.43 \overline{36} = \frac{4 3}{1 0 0} + \frac{1}{2 7 5}

Finally, we will add the fractions to obtain the value of

0.43 \overline{36}

expressed as a single fraction.

0.43 \overline{36} = \frac{4 3}{1 0 0} + \frac{1}{2 7 5}

▼

Simplify right-hand side

$\frac{a}{b} = \frac{a \cdot 1 1}{b \cdot 1 1}$

0.43 \overline{36} = \frac{4 7 3}{1 1 0 0} + \frac{1}{2 7 5}

$\frac{a}{b} = \frac{a \cdot 4}{b \cdot 4}$

0.43 \overline{36} = \frac{4 7 3}{1 1 0 0} + \frac{4}{1 1 0 0}

Add fractions

0.43 \overline{36} = \frac{4 7 7}{1 1 0 0}

Subchapter links

Check Your Understanding p.686

H.O.T. Problems p.688

Standardized Test Practice p.689

Spiral Review p.689

Skills Review p.689