Verified Answer :
- Multiply the fractions \(\frac{1}{8}\) and \(\frac{9}{11}\) by multiplying the numerators and denominators to get \(\frac{9}{88}\).
- Add the fractions \(\frac{5}{6}\) and \(\frac{4}{7}\) by using a common denominator of 42 to get \(\frac{35}{42} + \frac{24}{42} = \frac{59}{42}\).
- Final results: \(\left(\frac{1}{8}\right)\left(\frac{9}{11}\right) = \boxed{\frac{9}{88}}\) and \(\frac{5}{6} + \frac{4}{7} = \boxed{\frac{59}{42}}\).
Explanation
- Data and Problem Analysis
We are tasked with performing two basic arithmetic operations: the multiplication of two fractions and the addition of two other fractions. Specifically, we need to calculate \(\left(\frac{1}{8}\right)\left(\frac{9}{11}\right)\) and \(\frac{5}{6} + \frac{4}{7}\). Both operations involve fundamental rules of arithmetic with rational numbers. For multiplication, we multiply the numerators together and the denominators together. For addition, we find a common denominator and then add the resulting numerators.
- Multiplying the Fractions
To multiply the fractions \(\frac{1}{8}\) and \(\frac{9}{11}\), we follow the rule: \(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\).
Applying this:
\[\left(\frac{1}{8}\right)\left(\frac{9}{11}\right) = \frac{1 \cdot 9}{8 \cdot 11} = \frac{9}{88}\]
The fraction \(\frac{9}{88}\) cannot be simplified further as the greatest common divisor of 9 and 88 is 1.
- Adding the Fractions
To add the fractions \(\frac{5}{6}\) and \(\frac{4}{7}\), we need a common denominator. The least common multiple of 6 and 7 is \(6 \times 7 = 42\).
We convert each fraction:
\[\frac{5}{6} = \frac{5 \cdot 7}{6 \cdot 7} = \frac{35}{42}\]
\[\frac{4}{7} = \frac{4 \cdot 6}{7 \cdot 6} = \frac{24}{42}\]
Adding them together:
\[\frac{35}{42} + \frac{24}{42} = \frac{35 + 24}{42} = \frac{59}{42}\]
The fraction \(\frac{59}{42}\) is in its simplest form.
- Final Answer and Conclusion
We have performed the operations and obtained the results:
\[\left(\frac{1}{8}\right)\left(\frac{9}{11}\right) = \boxed{\frac{9}{88}}\]
\[\frac{5}{6} + \frac{4}{7} = \boxed{\frac{59}{42}}\]
Examples
Arithmetic with fractions is essential in daily life, such as in cooking where recipes often require adding parts of cups or other measurements. For instance, if you add \(\frac{1}{2}\) cup of flour and \(\frac{1}{3}\) cup of sugar, you are performing fraction addition. Understanding these operations ensures you can correctly scale recipes or combine ingredients accurately for any project.