I don't know how to convert infinite periodic decimal number $x=3,1(42)=3.1424242...$ to a fraction $\frac{a}{b}$ $a,b$ are integers. Need to find $a,b$
Infinite repeated decimals in a number
4 Answers
Equation $x = 3.1\overline{42}$ first multiply by $10$ we get
$10x = 31.\overline{42}$ now multiply by $100$ we have
$1000x = 3142.\overline{42}$ from third equation subtract second we get
$990x = 3111$
$x = \frac{3111}{990}=\frac{1037}{330}$
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0Note it is "multiplying" and "multiply". – 2012-11-15
There are several ways to do this. One way would be first to focus on the repeating part. So let $ x = 0.042424242\ldots = 0.0\overline{42}. $
Then you have $100x = 4.2424242\ldots = 4.2\overline{42}$. So $ 99x = 100x - x = 4.2424242\ldots - 0.042424242 = 4.2. $ So $ x = \frac{4.2}{99} = \frac{42}{990}. $
Now then you can probably find $3.1424242\ldots = 3.1 + x = \ldots$
Any repeating decimal can be written as a convergent geometric series. Here we have $3.1424242\ldots=3.1+{42\over 10^3}+{42\over 10^5}+{42\over 10^7}+\cdots$ $=3.1+{42\over 10^3}\Bigl(1+{1\over 100}+{1\over 100^2}+{1\over 100^3}+\cdots \Bigr)$ $=3.1+{42\over 10^3}{1\over 1-{1\over 100}}=3.1+{42\over 990}$
$0.0\overline{42} = 42 \cdot 0.0\overline{01} = \frac{42}{99} \cdot 0.0\overline{99} = \frac{42}{99} \cdot 0.1$