Solve the following mixed number addition problems: a) 3 rac78 + 5 rac34 b) 2 rac25 + 3 rac710 c) 7 rac310 + 8 rac16

Here are the solutions for adding the mixed numbers:

a) $3\frac{7}{8} + 5\frac{3}{4}$

Step 1: Add the whole numbers.

$$3 + 5 = 8$$

Step 2: Find a common denominator for the fractions $\frac{7}{8}$ and $\frac{3}{4}$. The least common denominator is 8.

$$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$$

So, the expression becomes:

$$8\frac{7}{8} + \frac{\boxed{6}}{8}$$

Step 3: Add the fractions.

$$\frac{7}{8} + \frac{6}{8} = \frac{7+6}{8} = \frac{13}{8}$$

Step 4: Convert the improper fraction to a mixed number.

$$\frac{13}{8} = 1\frac{5}{8}$$

Step 5: Add this to the whole number from Step 1.

$$8 + 1\frac{5}{8} = 9\frac{5}{8}$$

The final answer is \boxed{\text{9\frac{5}{8}}}.

b) $2\frac{2}{5} + 3\frac{7}{10}$

Step 1: Add the whole numbers.

$$2 + 3 = 5$$

Step 2: Find a common denominator for the fractions $\frac{2}{5}$ and $\frac{7}{10}$. The least common denominator is 10.

$$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$$

So, the expression becomes:

$$5\frac{\boxed{4}}{10} + \frac{7}{10}$$

Step 3: Add the fractions.

$$\frac{4}{10} + \frac{7}{10} = \frac{4+7}{10} = \frac{11}{10}$$

Step 4: Convert the improper fraction to a mixed number.

$$\frac{11}{10} = 1\frac{1}{10}$$

Step 5: Add this to the whole number from Step 1.

$$5 + 1\frac{1}{10} = 6\frac{1}{10}$$

The final answer is \boxed{\text{6\frac{1}{10}}}.

c) $7\frac{3}{10} + 8\frac{1}{6}$

Step 1: Add the whole numbers.

$$7 + 8 = 15$$

Step 2: Find a common denominator for the fractions $\frac{3}{10}$ and $\frac{1}{6}$. The least common multiple of 10 and 6 is 30.

$$\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}$$ $$\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}$$

Step 3: Add the fractions.

$$\frac{9}{30} + \frac{5}{30} = \frac{9+5}{30} = \frac{14}{30}$$

Step 4: Simplify the fraction.

$$\frac{14}{30} = \frac{14 \div 2}{30 \div 2} = \frac{7}{15}$$

Step 5: Combine the whole number and the simplified fraction.

$$15 + \frac{7}{15} = 15\frac{7}{15}$$

The final answer is \boxed{\text{15\frac{7}{15}}}.

d) $6\frac{4}{5} + \frac{5}{6}$

Step 1: The whole number is 6.

Step 2: Find a common denominator for the fractions $\frac{4}{5}$ and $\frac{5}{6}$. The least common multiple of 5 and 6 is 30.

$$\frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}$$ $$\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}$$

Step 3: Add the fractions.

$$\frac{24}{30} + \frac{25}{30} = \frac{24+25}{30} = \frac{49}{30}$$

Step 4: Convert the improper fraction to a mixed number.

$$\frac{49}{30} = 1\frac{19}{30}$$

Step 5: Add this to the whole number from Step 1.

$$6 + 1\frac{19}{30} = 7\frac{19}{30}$$

The final answer is \boxed{\text{7\frac{19}{30}}}.

e) $\frac{3}{4} + 4\frac{1}{6}$

Step 1: The whole number is 4.

Step 2: Find a common denominator for the fractions $\frac{3}{4}$ and $\frac{1}{6}$. The least common multiple of 4 and 6 is 12.

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