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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 12: Q. 340 (page 1222)

Write the repeating decimal as a fraction $0 . \overset{炉}{81}$ .

Short Answer

Expert verified

On writing as a fraction, we get, $\frac{9}{11}$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

$0 . \overset{炉}{81}$

Rewrite the given decimal showing the repeating as,

$= 0.8181818181 . . .$

Use place value to rewrite this as a sum,

$= 0.81 + 0.0081 + 0.000081 + . . .$

This is an infinite geometric series.

Then, $a_{1} = 0.81$

Find the common ratio,

$r = \frac{0.0081}{0.81} = 0.01$

02

Step 2. Find the required sum.

As $|r| < 1$ then sum of first nterms of geometric sequence, $S = \frac{a_{1}}{1 - r} . . . . . . (i)$

Substitute the values in equation (i),

$S = \frac{0.81}{1 - 0.01} = \frac{0.81}{0.99} = \frac{9}{11}$

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In the following exercises, evaluate.

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