- The expression \(\frac{7}{8} + \frac{4}{8}\) combines to \(\boxed{\frac{11}{8}}\).
- The expression \(\frac{17}{25} - \frac{9}{25}\) simplifies to \(\boxed{\frac{8}{25}}\).
- The expression \(\frac{63}{46} - \frac{81}{46}\) yields \(\frac{-18}{46}\), which simplifies to \(\boxed{-\frac{9}{23}}\).
Explanation
- Understanding the Method for Adding and Subtracting Fractions
We are given three expressions involving fractions with common denominators. To solve these, we use the basic property of adding or subtracting fractions with the same denominator:
\[\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}\]
where \(a\) and \(b\) are the numerators and \(d\) is the common denominator. This allows us to combine the numerators directly while keeping the denominator the same. Once we find the resulting fraction, we check if it can be simplified further by dividing both the numerator and the denominator by their greatest common divisor.
- Calculating the First Expression
For the first expression, \(\frac{7}{8} + \frac{4}{8}\), we have a common denominator of 8. We add the numerators:
\[\frac{7}{8} + \frac{4}{8} = \frac{7 + 4}{8} = \frac{11}{8}\]
This fraction is in its simplest form, as 11 and 8 have no common factors other than 1.
- Calculating the Second Expression
For the second expression, \(\frac{17}{25} - \frac{9}{25}\), the common denominator is 25. We subtract the numerator 9 from 17:
\[\frac{17}{25} - \frac{9}{25} = \frac{17 - 9}{25} = \frac{8}{25}\]
The fraction \(\frac{8}{25}\) is in its simplest form because 8 and 25 share no common divisors other than 1.
- Calculating the Third Expression
For the third expression, \(\frac{63}{46} - \frac{81}{46}\), the common denominator is 46. We subtract the numerator 81 from 63:
\[\frac{63}{46} - \frac{81}{46} = \frac{63 - 81}{46} = \frac{-18}{46}\]
To simplify this fraction, we identify the greatest common divisor of -18 and 46, which is 2:
\[\frac{-18 \div 2}{46 \div 2} = -\frac{9}{23}\]
Thus, the simplified result is \(-\frac{9}{23}\).
Examples
Understanding fractions is essential in daily life, such as when following a recipe that calls for precise measurements. For instance, if you have \(\frac{3}{4}\) cup of flour and add another \(\frac{1}{4}\) cup, you are performing fraction addition: \(\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1\) cup of flour in total. Mastering these operations helps you manage resources, calculate distances, and handle financial transactions accurately.