Verified Answer :

  • Divide \(13714\) by \(72\) using long division to find the quotient \(190\) and remainder \(34\).
  • Represent the result as a mixed number: \(190 \frac{34}{72}\).
  • Simplify the fractional part \(\frac{34}{72}\) by dividing both the numerator and denominator by their greatest common divisor, \(2\), resulting in \(\frac{17}{36}\).
  • Express the final result as \(\boxed{190 \frac{17}{36}}\).

Explanation

  1. Perform the Division

The objective is to perform the division \(13714 \div 72\) and express the result in its simplest form. We are dividing a four-digit dividend, \(13714\), by a two-digit divisor, \(72\). To find the result, we perform long division:

  1. Divide \(137\) by \(72\): \(137 \div 72 = 1\) with a remainder of \(137 - 72 = 65\).
  2. Bring down the next digit \(1\): the number becomes \(651\). Divide \(651 \div 72\): \(651 \div 72 = 9\) (since \(9 \times 72 = 648\)) with a remainder of \(651 - 648 = 3\).
  3. Bring down the final digit \(4\): the number becomes \(34\). Divide \(34 \div 72\): \(34 \div 72 = 0\) with a remainder of \(34\).

Thus, the integer part of the quotient is \(190\) and the remainder is \(34\). This can be written as \(190 \frac{34}{72}\).

  1. Simplify the Result

To simplify the fraction \(\frac{34}{72}\), we need to find the greatest common divisor (GCD) of \(34\) and \(72\).

  1. The factors of \(34\) are \(1, 2, 17, 34\).
  2. The factors of \(72\) are \(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\).

The largest common factor is \(2\). Dividing both the numerator and denominator by \(2\), we get:
\(\frac{34 \div 2}{72 \div 2} = \frac{17}{36}\).
Therefore, the simplified form of the result is \(190 \frac{17}{36}\). In decimal form, this is approximately \(190.47222...\).

  1. Final Answer

The result of the division \(13714 \div 72\) is \(\boxed{190 \frac{17}{36}}\).

Examples

When dividing a large inventory or budget into smaller, equal parts, knowing the exact remainder is vital. For example, if you have \$13,714 to distribute equally among 72 departments, each department receives exactly \(\$190 \frac{17}{36}\). Understanding how to simplify fractions ensures that you can precisely allocate resources without losing any value due to rounding errors.

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