Question

add.
(1) $3\frac{5}{7} + 2\frac{5}{14} =$
(2) $\frac{5}{6} + 2\frac{4}{15} =$
(3) $5\frac{5}{6} + \frac{1}{16} =$
(4) $7\frac{7}{10} + 2\frac{11}{20} =$
(5) $\frac{1}{14} + \frac{4}{21} =$
(6) $6\frac{3}{10} + 3\frac{9}{25} =$

Explanation:

Step1: Add whole numbers and fractions separately

Whole numbers: $3+2=5$; Fractions: $\frac{5}{7}+\frac{5}{14}=\frac{10}{14}+\frac{5}{14}=\frac{15}{14}=1\frac{1}{14}$
Total: $5+1\frac{1}{14}=6\frac{1}{14}$? Correction: $\frac{15}{14}=1+\frac{1}{14}$, so $5+1+\frac{1}{14}=6\frac{1}{14}$? Wait no: $\frac{5}{7}=\frac{10}{14}$, $\frac{10+5}{14}=\frac{15}{14}=1\frac{1}{14}$, yes, $3+2+1\frac{1}{14}=6\frac{1}{14}$? Wait original problem (1): $3\frac{5}{7}+2\frac{5}{14}= (3+2)+(\frac{10}{14}+\frac{5}{14})=5+\frac{15}{14}=5+1\frac{1}{14}=6\frac{1}{14}$ (typo earlier, fixed here).

Step2: Convert to common denominator (30)

$\frac{5}{6}=\frac{25}{30}$, $2\frac{4}{15}=2\frac{8}{30}$; Sum: $2+\frac{25+8}{30}=2\frac{33}{30}=2\frac{11}{10}=3\frac{1}{10}$? Correction: $\frac{25+8}{30}=\frac{33}{30}=\frac{11}{10}=1\frac{1}{10}$, so $2+1\frac{1}{10}=3\frac{1}{10}$? Wait no: $\frac{5}{6}+\2\frac{4}{15}=\frac{25}{30}+2\frac{8}{30}=2+\frac{33}{30}=2+1\frac{3}{30}=2+1\frac{1}{10}=3\frac{1}{10}$.

Step3: Common denominator (48)

$5\frac{5}{6}=5\frac{40}{48}$, $\frac{1}{16}=\frac{3}{48}$; Sum: $5+\frac{40+3}{48}=5\frac{43}{48}$.

Step4: Common denominator (20)

$7\frac{7}{10}=7\frac{14}{20}$, $2\frac{11}{20}=2\frac{11}{20}$; Sum: $(7+2)+(\frac{14+11}{20})=9+\frac{25}{20}=9+\frac{5}{4}=10\frac{1}{4}$.

Step5: Common denominator (42)

$\frac{1}{14}=\frac{3}{42}$, $\frac{4}{21}=\frac{8}{42}$; Sum: $\frac{3+8}{42}=\frac{11}{42}$.

Step6: Common denominator (50)

$6\frac{3}{10}=6\frac{15}{50}$, $3\frac{9}{25}=3\frac{18}{50}$; Sum: $(6+3)+(\frac{15+18}{50})=9+\frac{33}{50}=9\frac{33}{50}$.

Final corrected answers:
(1) $6\frac{1}{14}$
(2) $3\frac{1}{10}$
(3) $5\frac{43}{48}$
(4) $10\frac{1}{4}$
(5) $\frac{11}{42}$
(6) $9\frac{33}{50}$

Wait let's redo (1) carefully: $3\frac{5}{7}=3+\frac{5}{7}$, $2\frac{5}{14}=2+\frac{5}{14}$. Add whole parts: 5. Add fractions: $\frac{5}{7}=\frac{10}{14}$, so $\frac{10+5}{14}=\frac{15}{14}=1\frac{1}{14}$. Total: $5+1\frac{1}{14}=6\frac{1}{14}$ (correct).

(2): $\frac{5}{6}+\2\frac{4}{15}=\frac{25}{30}+2\frac{8}{30}=2+\frac{33}{30}=2+1\frac{3}{30}=2+1\frac{1}{10}=3\frac{1}{10}$ (correct).

(3): $5\frac{5}{6}+\frac{1}{16}=5+\frac{5}{6}+\frac{1}{16}=5+\frac{40}{48}+\frac{3}{48}=5\frac{43}{48}$ (correct).

(4): $7\frac{7}{10}+2\frac{11}{20}=7+2+\frac{14}{20}+\frac{11}{20}=9+\frac{25}{20}=9+1\frac{5}{20}=10\frac{1}{4}$ (correct).

(5): $\frac{1}{14}+\frac{4}{21}=\frac{3}{42}+\frac{8}{42}=\frac{11}{42}$ (correct).

(6): $6\frac{3}{10}+3\frac{9}{25}=6+3+\frac{15}{50}+\frac{18}{50}=9+\frac{33}{50}=9\frac{33}{50}$ (correct).

So the final answers are as above.

Step1: Add mixed numbers (1)

Whole parts: $3+2=5$; Fractions: $\frac{10}{14}+\frac{5}{14}=\frac{15}{14}=1\frac{1}{14}$; Total: $5+1\frac{1}{14}=6\frac{1}{14}$

Step2: Add fractions and mixed number (2)

$\frac{25}{30}+2\frac{8}{30}=2+\frac{33}{30}=3\frac{1}{10}$

Step3: Add mixed number and fraction (3)

$5\frac{40}{48}+\frac{3}{48}=5\frac{43}{48}$

Step4: Add mixed numbers (4)

Whole parts: $7+2=9$; Fractions: $\frac{14}{20}+\frac{11}{20}=\frac{25}{20}=1\frac{1}{4}$; Total: $9+1\frac{1}{4}=10\frac{1}{4}$

Step5: Add proper fractions (5)

$\frac{3}{42}+\frac{8}{42}=\frac{11}{42}$

Step6: Add mixed numbers (6)

Whole parts: $6+3=9$; Fractions: $\frac{15}{50}+\frac{18}{50}=\frac{33}{50}$; Total: $9\frac{33}{50}$

Answer:

$6\frac{3}{14}$
$2\frac{13}{15}$
$5\frac{43}{48}$
$9\frac{1}{4}$
$\frac{11}{42}$
$9\frac{27}{50}$