Sure, let's solve these fractional arithmetic problems step-by-step.
1. \(\frac{3}{4} + \frac{7}{4} = 2.5\)
These two fractions have the same denominator, so we can directly add their numerators:
[tex]\[
\frac{3 + 7}{4} = \frac{10}{4} = 2.5
\][/tex]
2. \(\frac{4}{3} + \frac{7}{5} = 2.7333333333333334\)
To add these fractions, we need a common denominator, which is 15:
[tex]\[
\frac{4}{3} + \frac{7}{5} = \frac{4 \times 5}{3 \times 5} + \frac{7 \times 3}{5 \times 3} = \frac{20}{15} + \frac{21}{15} = \frac{41}{15} \approx 2.7333333333333334
\][/tex]
3. \(\frac{5}{6} + \frac{7}{6} = 2.0\)
These two fractions have the same denominator, so we can directly add their numerators:
[tex]\[
\frac{5 + 7}{6} = \frac{12}{6} = 2.0
\][/tex]
4. \(\frac{7}{8} + \frac{3}{8} = 1.25\)
These two fractions have the same denominator, so we can directly add their numerators:
[tex]\[
\frac{7 + 3}{8} = \frac{10}{8} = 1.25
\][/tex]
5. \(\frac{5}{7} + \frac{3}{14} = 0.9285714285714286\)
To add these fractions, we need a common denominator, which is 14:
[tex]\[
\frac{5}{7} + \frac{3}{14} = \frac{5 \times 2}{7 \times 2} + \frac{3}{14} = \frac{10}{14} + \frac{3}{14} = \frac{13}{14} \approx 0.9285714285714286
\][/tex]
6. \(\frac{3}{10} - \frac{2}{5} = -0.10000000000000003\)
To subtract these fractions, we need a common denominator, which is 10:
[tex]\[
\frac{3}{10} - \frac{2 \times 2}{5 \times 2} = \frac{3}{10} - \frac{4}{10} = \frac{3 - 4}{10} = \frac{-1}{10} = -0.1
\][/tex]
7. \(\frac{2}{3} \times \frac{3}{4} = 0.5\)
Multiply the numerators and denominators:
[tex]\[
\frac{2 \times 3}{3 \times 4} = \frac{6}{12} = 0.5
\][/tex]
8. \(\frac{5}{6} \times \frac{2}{5} = 0.33333333333333337\)
Multiply the numerators and denominators:
[tex]\[
\frac{5 \times 2}{6 \times 5} = \frac{10}{30} = \frac{1}{3} = 0.33333333333333337
\][/tex]
9. \(\frac{3}{4} \div \frac{2}{5} = 1.875\)
To divide by a fraction, multiply by its reciprocal:
[tex]\[
\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} = 1.875
\][/tex]
10. \(\frac{5}{9} \times \frac{3}{7} = 0.2380952380952381\)
Multiply the numerators and denominators:
[tex]\[
\frac{5 \times 3}{9 \times 7} = \frac{15}{63} = \frac{5}{21} \approx 0.2380952380952381
\][/tex]
11. \(\frac{7}{8} \div \frac{1}{2} = 1.75\)
To divide by a fraction, multiply by its reciprocal:
[tex]\[
\frac{7}{8} \div \frac{1}{2} = \frac{7}{8} \times \frac{2}{1} = \frac{7 \times 2}{8 \times 1} = \frac{14}{8} = 1.75
\][/tex]
These are the detailed solutions to the fractional arithmetic problems.
