Verified Answer :
- Rewrite the subtraction of a negative value as an addition: \(21 \frac{5}{6} + 7 \frac{2}{3}\).
- Separate the mixed numbers into their integer and fractional components: \((21 + 7) + (\frac{5}{6} + \frac{2}{3})\).
- Find a common denominator to add the fractions: \(\frac{5}{6} + \frac{4}{6} = \frac{9}{6} = 1 \frac{1}{2}\).
- Combine the results: \(28 + 1 \frac{1}{2} = \boxed{29 \frac{1}{2}}\).
Explanation
- Analyze the Expression
We are asked to evaluate the expression \(21 \frac{5}{6} - (-7 \frac{2}{3})\). This problem involves the subtraction of a negative mixed fraction. Recalling the rule for subtraction, subtracting a negative number is equivalent to adding its positive counterpart. Therefore, the expression can be rewritten as \(21 \frac{5}{6} + 7 \frac{2}{3}\).
- Decompose and Group Terms
To simplify the calculation, we can decompose each mixed number into its integer part and its fractional part. The expression becomes:
\[(21 + \frac{5}{6}) + (7 + \frac{2}{3})\]
By the associative and commutative properties of addition, we can group the integer parts together and the fractional parts together:
\[(21 + 7) + (\frac{5}{6} + \frac{2}{3})\]
Calculating the sum of the integers, we get \(21 + 7 = 28\). Now, we focus on adding the fractions.
- Calculate the Sum of Fractions
To add \(\frac{5}{6}\) and \(\frac{2}{3}\), we must find a common denominator. The least common multiple of 6 and 3 is 6. We convert the second fraction:
\[\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\]
Now, add the two fractions:
\[\frac{5}{6} + \frac{4}{6} = \frac{9}{6}\]
We can simplify the fraction \(\frac{9}{6}\) by dividing the numerator and denominator by their greatest common divisor, which is 3:
\[\frac{9 \div 3}{6 \div 3} = \frac{3}{2}\]
Converting \(\frac{3}{2}\) back into a mixed number, we get \(\frac{3}{2} = 1 \frac{1}{2}\).
- Final Calculation
Finally, we add the sum of the integer parts to the sum of the fractional parts:
\[28 + 1 \frac{1}{2} = 29 \frac{1}{2}\]
Alternatively, in decimal form, this is \(29.5\). Therefore, the value of the expression is \(\boxed{29 \frac{1}{2}}\).
Examples
Calculating with fractions is essential in everyday life, such as in cooking or construction. For instance, if a recipe calls for \(21 \frac{5}{6}\) grams of a specific flour and you accidentally remove \(7 \frac{2}{3}\) grams more than intended from your pantry (represented as subtracting a negative amount), you can use this method to determine the total quantity you have effectively used or accounted for. Understanding how to handle mixed numbers and common denominators allows you to manage precise measurements efficiently.