Convert the following repeating decimal to a fraction in simplest form. .9 bar(6) Answer:

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Convert the following repeating decimal to a fraction in simplest form.  .9 bar(6) Answer:

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Denote Repeating Decimal: Let's denote the repeating decimal 0.9 with a repeating 6 as x. x = 0.966666... To convert this repeating decimal to a fraction, we can use an algebraic method where we multiply x by a power of 10 to move the decimal point to the right of the repeating digits.Multiply by 10: We multiply x by 10 to shift the decimal point one place to the right: 10x = 9.666666... Notice that the digits after the decimal point are the same as in x, which is crucial for the next step.Subtract to Eliminate Decimals: Now we set up an equation to subtract x from 10x, which will eliminate the repeating decimals: 10x - x = 9.666666... - 0.966666... This subtraction will give us a whole number on the right side of the equation.Solve for x: Performing the subtraction, we get: 9x = 8.7 Now we have an equation without repeating decimals that we can solve for x.Divide by 9: To find x, we divide both sides of the equation by 9: x = 8.7 / 9Convert to Fraction: Now we convert the decimal 8.7 to a fraction. The decimal 8.7 is the same as 87/10, so we substitute that into our equation: x = (87/10) / 9Simplify Fraction: To simplify the fraction, we multiply the denominator 10 by 9: x = 87 / (10 * 9) x = 87 / 90Find GCD and Divide: We can simplify the fraction 87/90 by finding the greatest common divisor (GCD) of 87 and 90 and dividing both the numerator and the denominator by the GCD. The GCD of 87 and 90 is 3.Final Simplified Fraction: Divide both the numerator and the denominator by the GCD: x = (87 ÷ 3) / (90 ÷ 3) x = 29 / 30 This is the fraction in its simplest form.

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.9 \overline{6}$ $\newline$ Answer:

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