.571428571 as a fraction

Converting repeating decimals to fractions can be a bit tricky, but it’s definitely doable. Let’s break down the process for converting 0.571428571… to a fraction.

Step 1: Identify the repeating part

In this decimal, 0.571428571…, the sequence “571428” repeats indefinitely.

Step 2: Set up an equation

Let $x$ represent the repeating decimal:

$x=0.571428571428\dots$

Step 3: Multiply to shift the decimal point

Since the repeating sequence has 6 digits, multiply both sides of the equation by 10610^6106 (which is 1,000,000) to move the decimal point six places to the right:

$1,000,000\times x=571,428.571428571428\dots$

Step 4: Subtract the original equation from this new equation

This subtraction will eliminate the repeating part:.571428571 as a fraction

1,000,000x=571,428.571428571428…−x=0.571428571428…999,999x=571,428\begin{align*} 1,000,000x &= 571,428.571428571428… \\ – x &= 0.571428571428… \\ \hline 999,999x &= 571,428 \end{align*}1,000,000x−x999,999x​=571,428.571428571428…=0.571428571428…=571,428​​

Step 5: Solve for $x$

Divide both sides by 999,999:

x=571,428999,999x = \frac{571,428}{999,999}x=999,999571,428​

Step 6: Simplify the fraction

To simplify, find the greatest common divisor (GCD) of 571,428 and 999,999. Both numbers are divisible by 142,857:

571,428÷142,857999,999÷142,857=47\begin{align*} \frac{571,428 \div 142,857}{999,999 \div 142,857} &= \frac{4}{7} \end{align*}999,999÷142,857571,428÷142,857​​=74​​

Therefore, 0.571428571428… as a fraction is 47\frac{4}{7}74​.

Verification

To ensure accuracy, divide 4 by 7:.571428571 as a fraction

4÷7=0.571428‾4 \div 7 = 0.\overline{571428}4÷7=0.571428

This confirms that the decimal representation of 47\frac{4}{7}74​ is indeed 0.571428571428…, verifying our conversion.

Understanding how to convert repeating decimals to fractions is a valuable skill in mathematics, enhancing comprehension of the relationship between different numerical representations.