Worksheet on operations with rational numbers, featuring definitions, examples, and practice exercises.

Problem Overview:

The image provides a summary of operations with rational numbers, including addition, subtraction, multiplication, division, and handling multiple operations. It includes examples and practice problems. Below, I will solve the practice problems step by step.

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Practice Problems:

#### Section 1: Evaluate (Addition and Subtraction)

1. (a) $\frac{1}{4} + \frac{-3}{4}$
- Since the denominators are the same, add the numerators directly:
$$
\frac{1}{4} + \frac{-3}{4} = \frac{1 + (-3)}{4} = \frac{-2}{4}
$$
- Simplify the fraction:
$$
\frac{-2}{4} = \frac{-1}{2}
$$
- Answer: $\boxed{-\frac{1}{2}}$

2. (b) $\frac{1}{2} - \frac{-2}{3}$
- Subtracting a negative is the same as adding:
$$
\frac{1}{2} - \frac{-2}{3} = \frac{1}{2} + \frac{2}{3}
$$
- Find a common denominator (LCM of 2 and 3 is 6):
$$
\frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}
$$
- Add the fractions:
$$
\frac{3}{6} + \frac{4}{6} = \frac{7}{6}
$$
- Answer: $\boxed{\frac{7}{6}}$

3. (c) $\frac{-3}{4} - \frac{1}{-4}$
- Simplify the second fraction:
$$
\frac{1}{-4} = -\frac{1}{4}
$$
- Rewrite the expression:
$$
\frac{-3}{4} - \left(-\frac{1}{4}\right) = \frac{-3}{4} + \frac{1}{4}
$$
- Add the fractions (same denominator):
$$
\frac{-3}{4} + \frac{1}{4} = \frac{-3 + 1}{4} = \frac{-2}{4}
$$
- Simplify the fraction:
$$
\frac{-2}{4} = \frac{-1}{2}
$$
- Answer: $\boxed{-\frac{1}{2}}$

4. (d) $\frac{-3}{5} + \frac{3}{-4}$
- Simplify the second fraction:
$$
\frac{3}{-4} = -\frac{3}{4}
$$
- Rewrite the expression:
$$
\frac{-3}{5} + \left(-\frac{3}{4}\right) = \frac{-3}{5} - \frac{3}{4}
$$
- Find a common denominator (LCM of 5 and 4 is 20):
$$
\frac{-3}{5} = \frac{-12}{20}, \quad \frac{3}{4} = \frac{15}{20}
$$
- Subtract the fractions:
$$
\frac{-12}{20} - \frac{15}{20} = \frac{-12 - 15}{20} = \frac{-27}{20}
$$
- Answer: $\boxed{-\frac{27}{20}}$

5. (e) $\frac{-1}{4} - 1\frac{1}{3}$
- Convert the mixed number to an improper fraction:
$$
1\frac{1}{3} = \frac{4}{3}
$$
- Rewrite the expression:
$$
\frac{-1}{4} - \frac{4}{3}
$$
- Find a common denominator (LCM of 4 and 3 is 12):
$$
\frac{-1}{4} = \frac{-3}{12}, \quad \frac{4}{3} = \frac{16}{12}
$$
- Subtract the fractions:
$$
\frac{-3}{12} - \frac{16}{12} = \frac{-3 - 16}{12} = \frac{-19}{12}
$$
- Answer: $\boxed{-\frac{19}{12}}$

6. (f) $-8\frac{1}{4} - \frac{-1}{-3}$
- Simplify the second fraction:
$$
\frac{-1}{-3} = \frac{1}{3}
$$
- Rewrite the expression:
$$
-8\frac{1}{4} - \frac{1}{3}
$$
- Convert the mixed number to an improper fraction:
$$
-8\frac{1}{4} = -\frac{33}{4}
$$
- Rewrite the expression:
$$
-\frac{33}{4} - \frac{1}{3}
$$
- Find a common denominator (LCM of 4 and 3 is 12):
$$
-\frac{33}{4} = -\frac{99}{12}, \quad \frac{1}{3} = \frac{4}{12}
$$
- Subtract the fractions:
$$
-\frac{99}{12} - \frac{4}{12} = \frac{-99 - 4}{12} = \frac{-103}{12}
$$
- Answer: $\boxed{-\frac{103}{12}}$

7. (g) $\frac{2}{-3} - 1\frac{5}{6}$
- Simplify the first fraction:
$$
\frac{2}{-3} = -\frac{2}{3}
$$
- Convert the mixed number to an improper fraction:
$$
1\frac{5}{6} = \frac{11}{6}
$$
- Rewrite the expression:
$$
-\frac{2}{3} - \frac{11}{6}
$$
- Find a common denominator (LCM of 3 and 6 is 6):
$$
-\frac{2}{3} = -\frac{4}{6}
$$
- Subtract the fractions:
$$
-\frac{4}{6} - \frac{11}{6} = \frac{-4 - 11}{6} = \frac{-15}{6}
$$
- Simplify the fraction:
$$
\frac{-15}{6} = \frac{-5}{2}
$$
- Answer: $\boxed{-\frac{5}{2}}$

8. (h) $\frac{5}{-6} - 2\frac{1}{3}$
- Simplify the first fraction:
$$
\frac{5}{-6} = -\frac{5}{6}
$$
- Convert the mixed number to an improper fraction:
$$
2\frac{1}{3} = \frac{7}{3}
$$
- Rewrite the expression:
$$
-\frac{5}{6} - \frac{7}{3}
$$
- Find a common denominator (LCM of 6 and 3 is 6):
$$
\frac{7}{3} = \frac{14}{6}
$$
- Subtract the fractions:
$$
-\frac{5}{6} - \frac{14}{6} = \frac{-5 - 14}{6} = \frac{-19}{6}
$$
- Answer: $\boxed{-\frac{19}{6}}$

9. (i) $\frac{-3}{5} + \frac{-3}{4} - \frac{7}{10}$
- Find a common denominator (LCM of 5, 4, and 10 is 20):
$$
\frac{-3}{5} = \frac{-12}{20}, \quad \frac{-3}{4} = \frac{-15}{20}, \quad \frac{7}{10} = \frac{14}{20}
$$
- Add and subtract the fractions:
$$
\frac{-12}{20} + \frac{-15}{20} - \frac{14}{20} = \frac{-12 - 15 - 14}{20} = \frac{-41}{20}
$$
- Answer: $\boxed{-\frac{41}{20}}$

10. (j) $\frac{2}{3} - \frac{-1}{2} - \frac{1}{-6}$
- Simplify the second and third fractions:
$$
\frac{-1}{2} = -\frac{1}{2}, \quad \frac{1}{-6} = -\frac{1}{6}
$$
- Rewrite the expression:
$$
\frac{2}{3} - \left(-\frac{1}{2}\right) - \left(-\frac{1}{6}\right) = \frac{2}{3} + \frac{1}{2} + \frac{1}{6}
$$
- Find a common denominator (LCM of 3, 2, and 6 is 6):
$$
\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{6} = \frac{1}{6}
$$
- Add the fractions:
$$
\frac{4}{6} + \frac{3}{6} + \frac{1}{6} = \frac{4 + 3 + 1}{6} = \frac{8}{6}
$$
- Simplify the fraction:
$$
\frac{8}{6} = \frac{4}{3}
$$
- Answer: $\boxed{\frac{4}{3}}$

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#### Section 2: Evaluate (Multiplication and Division)

1. (a) $\frac{4}{5} \times \frac{-20}{25}$
- Multiply the numerators and denominators:
$$
\frac{4}{5} \times \frac{-20}{25} = \frac{4 \times (-20)}{5 \times 25} = \frac{-80}{125}
$$
- Simplify the fraction:
$$
\frac{-80}{125} = \frac{-16}{25}
$$
- Answer: $\boxed{-\frac{16}{25}}$

2. (b) $\frac{-3}{2} \times \frac{6}{5}$
- Multiply the numerators and denominators:
$$
\frac{-3}{2} \times \frac{6}{5} = \frac{-3 \times 6}{2 \times 5} = \frac{-18}{10}
$$
- Simplify the fraction:
$$
\frac{-18}{10} = \frac{-9}{5}
$$
- Answer: $\boxed{-\frac{9}{5}}$

3. (c) $\left(\frac{-1}{3}\right)\left(\frac{2}{-5}\right)$
- Simplify the second fraction:
$$
\frac{2}{-5} = -\frac{2}{5}
$$
- Multiply the numerators and denominators:
$$
\left(\frac{-1}{3}\right)\left(-\frac{2}{5}\right) = \frac{-1 \times (-2)}{3 \times 5} = \frac{2}{15}
$$
- Answer: $\boxed{\frac{2}{15}}$

4. (d) $\left(\frac{9}{4}\right)\left(\frac{-2}{-3}\right)$
- Simplify the second fraction:
$$
\frac{-2}{-3} = \frac{2}{3}
$$
- Multiply the numerators and denominators:
$$
\left(\frac{9}{4}\right)\left(\frac{2}{3}\right) = \frac{9 \times 2}{4 \times 3} = \frac{18}{12}
$$
- Simplify the fraction:
$$
\frac{18}{12} = \frac{3}{2}
$$
- Answer: $\boxed{\frac{3}{2}}$

5. (e) $\left(\frac{1}{-2}\right)\left(\frac{-2}{5}\right)$
- Simplify the first fraction:
$$
\frac{1}{-2} = -\frac{1}{2}
$$
- Multiply the numerators and denominators:
$$
\left(-\frac{1}{2}\right)\left(\frac{-2}{5}\right) = \frac{-1 \times (-2)}{2 \times 5} = \frac{2}{10}
$$
- Simplify the fraction:
$$
\frac{2}{10} = \frac{1}{5}
$$
- Answer: $\boxed{\frac{1}{5}}$

6. (f) $\frac{-4}{5} \times \frac{10}{-4}$
- Multiply the numerators and denominators:
$$
\frac{-4}{5} \times \frac{10}{-4} = \frac{-4 \times 10}{5 \times (-4)} = \frac{-40}{-20}
$$
- Simplify the fraction:
$$
\frac{-40}{-20} = 2
$$
- Answer: $\boxed{2}$

7. (g) $\left(\frac{-5}{12}\right)(-24)$
- Multiply the numerator by -24:
$$
\left(\frac{-5}{12}\right)(-24) = \frac{-5 \times (-24)}{12} = \frac{120}{12}
$$
- Simplify the fraction:
$$
\frac{120}{12} = 10
$$
- Answer: $\boxed{10}$

8. (h) $\left(-2\frac{1}{4}\right)\left(\frac{2}{-9}\right)$
- Convert the mixed number to an improper fraction:
$$
-2\frac{1}{4} = -\frac{9}{4}
$$
- Simplify the second fraction:
$$
\frac{2}{-9} = -\frac{2}{9}
$$
- Multiply the fractions:
$$
\left(-\frac{9}{4}\right)\left(-\frac{2}{9}\right) = \frac{-9 \times (-2)}{4 \times 9} = \frac{18}{36}
$$
- Simplify the fraction:
$$
\frac{18}{36} = \frac{1}{2}
$$
- Answer: $\boxed{\frac{1}{2}}$

9. (i) $\left(-1\frac{1}{10}\right)\left(3\frac{1}{11}\right)$
- Convert both mixed numbers to improper fractions:
$$
-1\frac{1}{10} = -\frac{11}{10}, \quad 3\frac{1}{11} = \frac{34}{11}
$$
- Multiply the fractions:
$$
\left(-\frac{11}{10}\right)\left(\frac{34}{11}\right) = \frac{-11 \times 34}{10 \times 11} = \frac{-374}{110}
$$
- Simplify the fraction:
$$
\frac{-374}{110} = \frac{-187}{55}
$$
- Answer: $\boxed{-\frac{187}{55}}$

10. (j) $-4\frac{1}{6} \times -7\frac{3}{4}$
- Convert both mixed numbers to improper fractions:
$$
-4\frac{1}{6} = -\frac{25}{6}, \quad -7\frac{3}{4} = -\frac{31}{4}
$$
- Multiply the fractions:
$$
\left(-\frac{25}{6}\right)\left(-\frac{31}{4}\right) = \frac{-25 \times (-31)}{6 \times 4} = \frac{775}{24}
$$
- Answer: $\boxed{\frac{775}{24}}$

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Final Answers:

1. (a) $\boxed{-\frac{1}{2}}$
2. (b) $\boxed{\frac{7}{6}}$
3. (c) $\boxed{-\frac{1}{2}}$
4. (d) $\boxed{-\frac{27}{20}}$
5. (e) $\boxed{-\frac{19}{12}}$
6. (f) $\boxed{-\frac{103}{12}}$
7. (g) $\boxed{-\frac{5}{2}}$
8. (h) $\boxed{-\frac{19}{6}}$
9. (i) $\boxed{-\frac{41}{20}}$
10. (j) $\boxed{\frac{4}{3}}$

11. (a) $\boxed{-\frac{16}{25}}$
12. (b) $\boxed{-\frac{9}{5}}$
13. (c) $\boxed{\frac{2}{15}}$
14. (d) $\boxed{\frac{3}{2}}$
15. (e) $\boxed{\frac{1}{5}}$
16. (f) $\boxed{2}$
17. (g) $\boxed{10}$
18. (h) $\boxed{\frac{1}{2}}$
19. (i) $\boxed{-\frac{187}{55}}$
20. (j) $\boxed{\frac{775}{24}}$