How do you turn 6.6 repeating into a fraction
The Correct Answer and Explanation is :
To convert the repeating decimal ( 6.6\overline{6} ) into a fraction, we can follow a systematic approach. The repeating decimal can be expressed as ( 6.6666…), where the digit 6 repeats indefinitely. Here’s how to do it step by step:
- Define the Decimal: Let ( x = 6.6666…).
- Eliminate the Repeating Part: To isolate the repeating decimal, multiply ( x ) by 10. This shifts the decimal point one place to the right:
[
10x = 66.6666…
] - Set Up the Equation: Now, you have two equations:
[
x = 6.6666…
]
[
10x = 66.6666…
] - Subtract the First Equation from the Second: Subtract the first equation from the second to eliminate the repeating part:
[
10x – x = 66.6666… – 6.6666…
]
This simplifies to:
[
9x = 60
] - Solve for ( x ): Divide both sides by 9 to isolate ( x ):
[
x = \frac{60}{9}
] - Simplify the Fraction: To simplify ( \frac{60}{9} ), divide both the numerator and denominator by their greatest common divisor (GCD), which is 3:
[
x = \frac{60 \div 3}{9 \div 3} = \frac{20}{3}
] - Final Result: Thus, the repeating decimal ( 6.6\overline{6} ) can be expressed as the fraction ( \frac{20}{3} ).
Conclusion
The fraction ( \frac{20}{3} ) represents the repeating decimal ( 6.6\overline{6} ). This process demonstrates a fundamental method of converting repeating decimals into fractions by establishing an equation, eliminating the repeat through multiplication, and then solving for the unknown. Understanding this method is crucial for working with decimals in mathematics, particularly in situations where precise values are required, such as in calculations in finance, engineering, or physics.