Fraction Problems and Solutions

Problem 1:

Find the sum of the fractions:

\[ \frac{5}{8} + \frac{3}{4} = ? \]

Solution:

Convert \(\frac{3}{4}\) to an equivalent fraction with a denominator of 8: \(\frac{6}{8}\). Now, add:

\[ \frac{5}{8} + \frac{6}{8} = \frac{11}{8} = 1 \frac{3}{8} \]

Problem 2:

Simplify the fraction:

\[ \frac{56}{98} \]

Solution:

Find the greatest common divisor (GCD) of 56 and 98, which is 14. Divide the numerator and denominator by 14:

\[ \frac{56 \div 14}{98 \div 14} = \frac{4}{7} \]

Problem 3:

Solve for \( x \):

\[ \frac{3}{x} = \frac{9}{12} \]

Solution:

Cross-multiply to get:

\[ 3 \cdot 12 = 9 \cdot x \Rightarrow x = \frac{36}{9} = 4 \]

Problem 4:

Subtract the fractions:

\[ \frac{7}{10} - \frac{3}{5} = ? \]

Solution:

Convert \(\frac{3}{5}\) to an equivalent fraction with a denominator of 10: \(\frac{6}{10}\). Now, subtract:

\[ \frac{7}{10} - \frac{6}{10} = \frac{1}{10} \]

Problem 5:

Multiply the fractions:

\[ \frac{7}{12} \cdot \frac{3}{14} = ? \]

Solution:

Multiply the numerators and denominators:

\[ \frac{7 \cdot 3}{12 \cdot 14} = \frac{21}{168} = \frac{1}{8} \]

Problem 6:

Divide the fractions:

\[ \frac{5}{6} \div \frac{10}{9} = ? \]

Solution:

To divide, multiply by the reciprocal of the second fraction:

\[ \frac{5}{6} \times \frac{9}{10} = \frac{45}{60} = \frac{3}{4} \]

Problem 7:

If \(\frac{3}{x} = \frac{4}{5}\), solve for \(x\).

Solution:

Cross-multiply to get:

\[ 3 \cdot 5 = 4 \cdot x \Rightarrow x = \frac{15}{4} = 3.75 \]

Problem 8:

Find the reciprocal of \(\frac{9}{7}\) and simplify the result.

Solution:

The reciprocal of \(\frac{9}{7}\) is \(\frac{7}{9}\).

Problem 9:

Combine the mixed numbers and simplify:

\[ 1 \frac{2}{3} + 2 \frac{1}{6} = ? \]

Solution:

Convert to improper fractions:

\[ 1 \frac{2}{3} = \frac{5}{3} \quad \text{and} \quad 2 \frac{1}{6} = \frac{13}{6} \]

Find a common denominator and add:

\[ \frac{5 \cdot 2}{3 \cdot 2} + \frac{13}{6} = \frac{10}{6} + \frac{13}{6} = \frac{23}{6} = 3 \frac{5}{6} \]

Problem 10:

Simplify the complex fraction:

\[ \frac{\frac{3}{4}}{\frac{5}{6}} = ? \]

Solution:

To simplify, multiply by the reciprocal of the denominator:

\[ \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10} \]