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Convert the following repeating decimal to a fraction in simplest form. .3 bar(7) Answer:

Convert the following repeating decimal to a fraction in simplest form.\newline.37‾ .3 \overline{7} \newlineAnswer:

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Full solution

Q. Convert the following repeating decimal to a fraction in simplest form.\newline.37‾ .3 \overline{7} \newlineAnswer:
  1. Assign xx as decimal: Let xx equal the repeating decimal 0.370.37 with the 77 repeating indefinitely, so x=0.3777…x = 0.3777\ldots
  2. Multiply xx by 1010: To isolate the repeating part, we multiply xx by 1010, since there is one digit in the repeating cycle. This gives us 10x=3.7777…10x = 3.7777\ldots
  3. Subtract xx from 10x10x: Now we subtract the original xx from 10x10x to get rid of the repeating part: 10x−x=3.7777...−0.3777...10x - x = 3.7777... - 0.3777...
  4. Solve for x: Performing the subtraction, we get 9x=3.49x = 3.4 because the infinite repeating 77s cancel each other out.
  5. Express as fraction: Now we solve for xx by dividing both sides of the equation by 99: x=3.49x = \frac{3.4}{9}
  6. Simplify fraction: To express 3.43.4 as a fraction, we write it as 3410\frac{34}{10} because moving the decimal point two places to the right converts it to an integer.
  7. Find GCD: Now we have x=3410/9x = \frac{34}{10} / 9. To simplify this, we multiply the numerator by the reciprocal of the denominator: x=3410×19x = \frac{34}{10} \times \frac{1}{9}
  8. Divide by GCD: Multiplying the fractions, we get x=34(10×9)=3490x = \frac{34}{(10 \times 9)} = \frac{34}{90}
  9. Divide by GCD: Multiplying the fractions, we get x=34(10×9)=3490x = \frac{34}{(10 \times 9)} = \frac{34}{90} We simplify the fraction 3490\frac{34}{90} by finding the greatest common divisor (GCD) of 3434 and 9090, which is 22.
  10. Divide by GCD: Multiplying the fractions, we get x=34(10×9)=3490x = \frac{34}{(10 \times 9)} = \frac{34}{90} We simplify the fraction 3490\frac{34}{90} by finding the greatest common divisor (GCD) of 3434 and 9090, which is 22. Dividing both the numerator and the denominator by the GCD, we get x=(34/2)(90/2)=1745x = \frac{(34/2)}{(90/2)} = \frac{17}{45}

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