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What number must be subtracted from 67345643 to get 54903891 ?

22. What number must be subtracted from 6734564367345643 to get 54903891 54903891 ?

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Full solution

Q. 22. What number must be subtracted from 6734564367345643 to get 54903891 54903891 ?
  1. Identify Meaning of Bar Notation: Identify the meaning of the bar notation. The bar over the number 49038914903891 indicates that the digits under the bar are repeating. So, 54903891‾5\overline{4903891} represents the number 5.49038913901389139013891…5.49038913901389139013891\ldots and so on, with the sequence 49038914903891 repeating indefinitely.
  2. Convert to Fraction: Recognize that to subtract and find the original number, we need to work with a finite representation of the repeating decimal. Let's convert 54903891‾5\overline{4903891} to a fraction. To do this, let x=5.490389139013891…x = 5.490389139013891\ldots, then multiply xx by 10710^7 (which is 1000000010000000) to shift the decimal point 77 places to the right, getting 107x=54903891.4903891…10^7x = 54903891.4903891\ldots
  3. Subtract Original xx: Subtract the original xx from the 107x10^7x to get rid of the repeating part. This gives us 107x−x=54903891.4903891...−5.4903891...=5490388610^7x - x = 54903891.4903891... - 5.4903891... = 54903886.
  4. Simplify Left Side: Now, simplify the left side of the equation: 107x−x=(107−1)x=9999999x10^7x - x = (10^7 - 1)x = 9999999x.
  5. Solve for x: Solve for x by dividing both sides of the equation by 99999999999999: x = rac{54903886}{9999999}.
  6. Perform Division: Perform the division to find the value of xx: x=549038869999999=5+49038919999999x = \frac{54903886}{9999999} = 5 + \frac{4903891}{9999999}.
  7. Subtract Fraction from Whole Number: Now that we have the repeating decimal as a fraction, we can subtract it from 6734564367345643. Let's represent the fraction as FF for simplicity: F=49038919999999F = \frac{4903891}{9999999}. The subtraction we want to perform is 67345643−F67345643 - F.
  8. Substitute Fraction: Substitute the fraction back into the subtraction: 67345643−49038919999999.67345643 - \frac{4903891}{9999999}.
  9. Write Whole Number as Fraction: To subtract the fraction from the whole number, we can write the whole number as a fraction with the same denominator: 67345643×99999999999999−49038919999999\frac{67345643 \times 9999999}{9999999} - \frac{4903891}{9999999}.
  10. Perform Subtraction of Numerators: Perform the subtraction of the numerators while keeping the common denominator: [(67345643×9999999)−4903891]/9999999[(67345643 \times 9999999) - 4903891] / 9999999.
  11. Calculate Numerator: Calculate the numerator: (67345643×9999999)−4903891=673456429999999−4903891(67345643 \times 9999999) - 4903891 = 673456429999999 - 4903891.
  12. Perform Subtraction in Numerator: Perform the subtraction in the numerator: 673456429999999−4903891=673456425000108673456429999999 - 4903891 = 673456425000108.
  13. Fraction Representing Number: Now we have the fraction representing the number to be subtracted from 6734564367345643: (673456425000108)/9999999(673456425000108) / 9999999.
  14. Convert Fraction to Decimal: Since we are looking for a whole number to subtract, we need to convert this fraction back into a decimal to see if it represents a whole number. However, we can observe that the numerator is not a multiple of the denominator, which means the fraction does not represent a whole number. This indicates a math error has occurred in the previous steps.

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