1.66666666667 as a fraction in simplest form

1.66666666667 as a fraction in simplest form

The Correct Answer and Explanation is:

The decimal ( 1.66666666667 ) is a repeating decimal, often written as ( 1.\overline{6} ), where the 6 repeats indefinitely. To convert this repeating decimal into a fraction in its simplest form, we need to follow a series of steps.

Step 1: Set the decimal equal to a variable

Let ( x = 1.\overline{6} ), meaning that ( x ) represents the repeating decimal.

Step 2: Eliminate the repeating part

To eliminate the repeating part, multiply both sides of the equation by 10. This shifts the decimal point one place to the right:
[
10x = 16.\overline{6}
]
Now we have two equations:

1. ( x = 1.\overline{6} )
2. ( 10x = 16.\overline{6} )

Step 3: Subtract the equations

Next, subtract the first equation from the second:
[
10x – x = 16.\overline{6} – 1.\overline{6}
]
This simplifies to:
[
9x = 15
]
The repeating decimals cancel out.

Step 4: Solve for ( x )

Now, solve for ( x ) by dividing both sides of the equation by 9:
[
x = \frac{15}{9}
]

Step 5: Simplify the fraction

The fraction ( \frac{15}{9} ) can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:
[
x = \frac{15 \div 3}{9 \div 3} = \frac{5}{3}
]

Thus, the decimal ( 1.\overline{6} ) (or ( 1.66666666667 )) as a fraction in simplest form is ( \frac{5}{3} ).

Conclusion

The decimal ( 1.\overline{6} ) is equivalent to the fraction ( \frac{5}{3} ). This process demonstrates how to convert repeating decimals to fractions by isolating the repeating part, subtracting equations, and simplifying the result.