Let's go through each problem step by step. These problems involve estimating and then finding the sum.

1. $4 \frac{4}{5} + \left(-3 \frac{3}{5}\right)$

Estimate:

• $4 \frac{4}{5} \approx 5$
• $-3 \frac{3}{5} \approx -4$ Estimated sum: $5 + (-4) = 1$

Exact calculation:

• Convert mixed numbers to improper fractions: $4 \frac{4}{5} = \frac{24}{5}, \quad -3 \frac{3}{5} = -\frac{18}{5}$
• Find the sum: $\frac{24}{5} + \left(-\frac{18}{5}\right) = \frac{24 - 18}{5} = \frac{6}{5} = 1 \frac{1}{5}$

2. $-5 \frac{1}{8} + 1 \frac{3}{4}$

Estimate:

• $-5 \frac{1}{8} \approx -5$
• $1 \frac{3}{4} \approx 2$ Estimated sum: $-5 + 2 = -3$

Exact calculation:

• Convert mixed numbers to improper fractions: $-5 \frac{1}{8} = -\frac{41}{8}, \quad 1 \frac{3}{4} = \frac{7}{4} = \frac{14}{8}$
• Find the sum: $-\frac{41}{8} + \frac{14}{8} = \frac{-41 + 14}{8} = \frac{-27}{8} = -3 \frac{3}{8}$

3. $-13.7 + (-2.05)$

Estimate:

• $-13.7 \approx -14$
• $-2.05 \approx -2$ Estimated sum: $-14 + (-2) = -16$

Exact calculation:

• Find the sum: $-13.7 + (-2.05) = -15.75$

4. $-10 \frac{2}{3} + 4 \frac{1}{6}$

Estimate:

• $-10 \frac{2}{3} \approx -11$
• $4 \frac{1}{6} \approx 4$ Estimated sum: $-11 + 4 = -7$

Exact calculation:

• Convert mixed numbers to improper fractions: $-10 \frac{2}{3} = -\frac{32}{3}, \quad 4 \frac{1}{6} = \frac{25}{6}$
• Find the common denominator (6): $-\frac{32}{3} = -\frac{64}{6}, \quad \frac{25}{6}$
• Find the sum: $-\frac{64}{6} + \frac{25}{6} = \frac{-64 + 25}{6} = \frac{-39}{6} = -6 \frac{3}{6} = -6 \frac{1}{2}$

5. $-20 \frac{1}{3} + \left(-2 \frac{1}{4}\right)$

Estimate:

• $-20 \frac{1}{3} \approx -20$
• $-2 \frac{1}{4} \approx -2$ Estimated sum: $-20 + (-2) = -22$

Exact calculation:

• Convert mixed numbers to improper fractions: $-20 \frac{1}{3} = -\frac{61}{3}, \quad -2 \frac{1}{4} = -\frac{9}{4}$
• Find the common denominator (12): $-\frac{61}{3} = -\frac{244}{12}, \quad -\frac{9}{4} = -\frac{27}{12}$
• Find the sum: $-\frac{244}{12} + \left(-\frac{27}{12}\right) = \frac{-244 - 27}{12} = \frac{-271}{12} = -22 \frac{7}{12}$

Do you want more details or have any questions?

Related questions:

1. How can you estimate sums more accurately?
2. How do you convert a mixed number to an improper fraction?
3. What steps can you take to ensure fractions have common denominators before adding?
4. Why is it important to estimate before finding the exact sum?
5. What is the difference between adding positive and negative numbers?

Tip: When working with fractions, always simplify the result if possible to make your final answer easier to interpret.