Question

due sep 12, 11:59 pm · mp1
hw #1 - repeating decimals to fractions
0/100
add class comment
convert the following repeating decimals to a fraction.
0.6666...
0.32323232...
0.42424242...
0.072072072...
show your work on loose leaf paper
attachments
write a repeating decimal as a fraction

Explanation:

Step1: Let \(x = 0.666\cdots\)

\(x=0.666\cdots\), then \(10x = 6.666\cdots\). Subtract \(x\) from \(10x\): \(10x - x=6.666\cdots - 0.666\cdots\), \(9x = 6\), so \(x=\frac{6}{9}=\frac{2}{3}\)

Step2: Let \(x = 0.323232\cdots\)

Let \(x = 0.323232\cdots\), then \(100x=32.323232\cdots\). Subtract \(x\) from \(100x\): \(100x - x = 32.3232\cdots-0.3232\cdots\), \(99x = 32\), so \(x=\frac{32}{99}\)

Step3: Let \(x = 0.424242\cdots\)

Let \(x = 0.424242\cdots\), then \(100x = 42.424242\cdots\). Subtract \(x\) from \(100x\): \(100x - x=42.4242\cdots - 0.4242\cdots\), \(99x = 42\), so \(x=\frac{42}{99}=\frac{14}{33}\)

Step4: Let \(x = 0.072072072\cdots\)

Let \(x = 0.072072\cdots\), then \(1000x=72.072072\cdots\). Subtract \(x\) from \(1000x\): \(1000x - x=72.072072\cdots - 0.072072\cdots\), \(999x = 72\), so \(x=\frac{72}{999}=\frac{8}{111}\)

Answer:

\(0.666\cdots=\frac{2}{3}\), \(0.323232\cdots=\frac{32}{99}\), \(0.424242\cdots=\frac{14}{33}\), \(0.072072072\cdots=\frac{8}{111}\)