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Convert the following repeating decimal to a fraction in simplest form. .2 bar(3) Answer:

Convert the following repeating decimal to a fraction in simplest form.\newline.23‾ .2 \overline{3} \newlineAnswer:

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Full solution

Q. Convert the following repeating decimal to a fraction in simplest form.\newline.23‾ .2 \overline{3} \newlineAnswer:
  1. Denote Repeating Decimal as xx: Let's denote the repeating decimal 0.20.2 with a repeating 33 as xx.\newlinex=0.23‾x = 0.2\overline{3}
  2. Multiply by 1010: To convert a repeating decimal to a fraction, we can set up an equation where the repeating part is isolated on one side. Since the 33 is the repeating part, we want to manipulate the equation to have only the repeating part on one side. To do this, we can multiply xx by 1010 to shift the decimal point to the right.\newline10x=2.3333…10x = 2.3333\ldots
  3. Subtract Original xx from 10x10x: Now, we subtract the original xx from 10x10x to get rid of the repeating part.\newline10x−x=2.3333...−0.2333...10x - x = 2.3333... - 0.2333...\newlineThis subtraction will leave us with 9x9x on the left side and the non-repeating part on the right side.\newline9x=2.19x = 2.1
  4. Solve for x: Now, we solve for x by dividing both sides of the equation by 99. \newlinex=2.19x = \frac{2.1}{9}
  5. Express 22.11 as Fraction: To express 2.12.1 as a fraction, we recognize that 2.12.1 is the same as 2110\frac{21}{10}. \newlinex=21109x = \frac{\frac{21}{10}}{9}
  6. Simplify by Multiplying Numerators: We can simplify this by multiplying the numerator by the reciprocal of the denominator. x=2110×19x = \frac{21}{10} \times \frac{1}{9}
  7. Find GCD and Simplify: Now, we multiply the numerators and the denominators. x=2190x = \frac{21}{90}
  8. Find GCD and Simplify: Now, we multiply the numerators and the denominators.\newlinex=2190x = \frac{21}{90}Finally, we simplify the fraction by finding the greatest common divisor (GCD) of 2121 and 9090, which is 33.\newlinex=(21÷3)(90÷3)x = \frac{(21 \div 3)}{(90 \div 3)}\newlinex=730x = \frac{7}{30}

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