Math worksheet detailing operations with rational numbers, including definitions, rules, examples, and practice exercises.

Let's solve the practice problems from your image step by step, following the rules of operations with rational numbers. We'll go through each part carefully.

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1. Evaluate

#### (a) $\frac{1}{4} + \frac{-3}{4}$

Same denominator:

$$
\frac{1 + (-3)}{4} = \frac{-2}{4} = -\frac{1}{2}
$$

✔ Answer: $-\frac{1}{2}$

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#### (b) $\frac{1}{2} - \frac{-2}{3}$

Subtracting a negative is adding:

$$
\frac{1}{2} + \frac{2}{3}
$$

Find common denominator (LCM of 2 and 3 is 6):

$$
= \frac{3}{6} + \frac{4}{6} = \frac{7}{6} = 1\frac{1}{6}
$$

✔ Answer: $\frac{7}{6}$ or $1\frac{1}{6}$

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#### (c) $\frac{-3}{4} - \frac{1}{-4}$

Note: $\frac{1}{-4} = -\frac{1}{4}$, so:

$$
\frac{-3}{4} - (-\frac{1}{4}) = \frac{-3}{4} + \frac{1}{4} = \frac{-2}{4} = -\frac{1}{2}
$$

✔ Answer: $-\frac{1}{2}$

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#### (d) $\frac{-3}{5} + \frac{3}{-4}$

$\frac{3}{-4} = -\frac{3}{4}$

So:

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

Common denominator: 20

$$
= -\frac{12}{20} - \frac{15}{20} = -\frac{27}{20} = -1\frac{7}{20}
$$

✔ Answer: $-\frac{27}{20}$ or $-1\frac{7}{20}$

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#### (e) $\frac{-1}{4} - 1\frac{1}{3}$

Convert mixed number: $1\frac{1}{3} = \frac{4}{3}$

So:

$$
\frac{-1}{4} - \frac{4}{3}
$$

Common denominator: 12

$$
= -\frac{3}{12} - \frac{16}{12} = -\frac{19}{12} = -1\frac{7}{12}
$$

✔ Answer: $-\frac{19}{12}$ or $-1\frac{7}{12}$

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#### (f) $-8\frac{1}{4} - \frac{-1}{-3}$

First simplify: $\frac{-1}{-3} = \frac{1}{3}$

Now: $-8\frac{1}{4} - \frac{1}{3}$

Convert $-8\frac{1}{4} = -\frac{33}{4}$

So:

$$
-\frac{33}{4} - \frac{1}{3}
$$

Common denominator: 12

$$
= -\frac{99}{12} - \frac{4}{12} = -\frac{103}{12} = -8\frac{7}{12}
$$

✔ Answer: $-\frac{103}{12}$ or $-8\frac{7}{12}$

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#### (g) $\frac{2}{-3} - 1\frac{5}{6}$

$\frac{2}{-3} = -\frac{2}{3}$

$1\frac{5}{6} = \frac{11}{6}$

So:

$$
-\frac{2}{3} - \frac{11}{6}
$$

Common denominator: 6

$$
= -\frac{4}{6} - \frac{11}{6} = -\frac{15}{6} = -\frac{5}{2} = -2\frac{1}{2}
$$

✔ Answer: $-\frac{5}{2}$ or $-2\frac{1}{2}$

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#### (h) $\frac{5}{-6} - 2\frac{1}{3}$

$\frac{5}{-6} = -\frac{5}{6}$

$2\frac{1}{3} = \frac{7}{3}$

So:

$$
-\frac{5}{6} - \frac{7}{3}
$$

Common denominator: 6

$$
= -\frac{5}{6} - \frac{14}{6} = -\frac{19}{6} = -3\frac{1}{6}
$$

✔ Answer: $-\frac{19}{6}$ or $-3\frac{1}{6}$

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#### (i) $\frac{-3}{5} + \frac{-3}{4} - \frac{7}{10}$

All are negative except possibly signs.

Add all three:

First: $\frac{-3}{5} + \frac{-3}{4} = -\frac{3}{5} - \frac{3}{4}$

Common denominator: 20

$$
= -\frac{12}{20} - \frac{15}{20} = -\frac{27}{20}
$$

Now subtract $\frac{7}{10}$:

$$
-\frac{27}{20} - \frac{7}{10} = -\frac{27}{20} - \frac{14}{20} = -\frac{41}{20} = -2\frac{1}{20}
$$

✔ Answer: $-\frac{41}{20}$ or $-2\frac{1}{20}$

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#### (j) $\frac{2}{3} - \frac{-1}{2} - \frac{1}{-6}$

Simplify signs:

$\frac{-1}{2} = -\frac{1}{2}$ → subtracting it becomes $+\frac{1}{2}$

$\frac{1}{-6} = -\frac{1}{6}$ → subtracting it becomes $+ \frac{1}{6}$

So expression becomes:

$$
\frac{2}{3} + \frac{1}{2} + \frac{1}{6}
$$

Common denominator: 6

$$
= \frac{4}{6} + \frac{3}{6} + \frac{1}{6} = \frac{8}{6} = \frac{4}{3} = 1\frac{1}{3}
$$

✔ Answer: $\frac{4}{3}$ or $1\frac{1}{3}$

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2. Evaluate (Multiplication)

#### (a) $\frac{4}{5} \times \frac{-20}{25}$

Simplify first:

$\frac{-20}{25} = -\frac{4}{5}$

So:

$$
\frac{4}{5} \times \left(-\frac{4}{5}\right) = -\frac{16}{25}
$$

✔ Answer: $-\frac{16}{25}$

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#### (b) $\frac{3}{-2} \times \frac{6}{5}$

$\frac{3}{-2} = -\frac{3}{2}$

$$
-\frac{3}{2} \times \frac{6}{5} = -\frac{18}{10} = -\frac{9}{5} = -1\frac{4}{5}
$$

✔ Answer: $-\frac{9}{5}$ or $-1\frac{4}{5}$

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#### (c) $\left(\frac{-1}{3}\right)\left(\frac{-2}{-5}\right)$

First: $\frac{-2}{-5} = \frac{2}{5}$

Now: $\frac{-1}{3} \times \frac{2}{5} = -\frac{2}{15}$

✔ Answer: $-\frac{2}{15}$

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#### (d) $\left(\frac{9}{4}\right)\left(\frac{-2}{-3}\right)$

$\frac{-2}{-3} = \frac{2}{3}$

So:

$$
\frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = \frac{3}{2} = 1\frac{1}{2}
$$

✔ Answer: $\frac{3}{2}$ or $1\frac{1}{2}$

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#### (e) $\left(\frac{1}{-2}\right)\left(\frac{-2}{5}\right)$

$\frac{1}{-2} = -\frac{1}{2}$

$$
-\frac{1}{2} \times \left(-\frac{2}{5}\right) = \frac{2}{10} = \frac{1}{5}
$$

✔ Answer: $\frac{1}{5}$

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#### (f) $\frac{-4}{5} \times \frac{10}{-4}$

Note: $\frac{10}{-4} = -\frac{5}{2}$

But better to cancel:

$$
\frac{-4}{5} \times \frac{10}{-4} = \frac{-4 \cdot 10}{5 \cdot (-4)} = \frac{-40}{-20} = 2
$$

Or cancel directly: $-4$ cancels, $10/5 = 2$, negatives cancel → $+2$

✔ Answer: $2$

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#### (g) $\left(\frac{-5}{12}\right)(-24)$

Multiply:

$$
\frac{-5}{12} \times (-24) = \frac{5}{12} \times 24 = \frac{5 \times 24}{12} = 5 \times 2 = 10
$$

✔ Answer: $10$

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#### (h) $\left(-2\frac{1}{4}\right)\left(\frac{2}{-9}\right)$

Convert: $-2\frac{1}{4} = -\frac{9}{4}$

$\frac{2}{-9} = -\frac{2}{9}$

Now:

$$
-\frac{9}{4} \times \left(-\frac{2}{9}\right) = \frac{18}{36} = \frac{1}{2}
$$

✔ Answer: $\frac{1}{2}$

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#### (i) $\left(-1\frac{1}{10}\right)\left(3\frac{1}{11}\right)$

Convert:

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

$3\frac{1}{11} = \frac{34}{11}$

Now:

$$
-\frac{11}{10} \times \frac{34}{11} = -\frac{374}{110}
$$

Simplify: divide numerator and denominator by 2:

$$
= -\frac{187}{55}
$$

Check if reducible: 187 ÷ 11 = 17, 55 ÷ 11 = 5 → $-\frac{17}{5} = -3\frac{2}{5}$

✔ Answer: $-\frac{17}{5}$ or $-3\frac{2}{5}$

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#### (j) $-4\frac{1}{6} \times -7\frac{3}{4}$

Convert:

$-4\frac{1}{6} = -\frac{25}{6}$

$-7\frac{3}{4} = -\frac{31}{4}$

Now:

$$
\left(-\frac{25}{6}\right) \times \left(-\frac{31}{4}\right) = \frac{775}{24}
$$

Now simplify: $775 \div 24$

Divide: $24 \times 32 = 768$, remainder 7 → $32\frac{7}{24}$

✔ Answer: $\frac{775}{24}$ or $32\frac{7}{24}$

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✔ Final Answers Summary

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

#### 2. Evaluate
(a) $-\frac{16}{25}$
(b) $-\frac{9}{5}$ or $-1\frac{4}{5}$
(c) $-\frac{2}{15}$
(d) $\frac{3}{2}$ or $1\frac{1}{2}$
(e) $\frac{1}{5}$
(f) $2$
(g) $10$
(h) $\frac{1}{2}$
(i) $-\frac{17}{5}$ or $-3\frac{2}{5}$
(j) $\frac{775}{24}$ or $32\frac{7}{24}$

Let me know if you'd like these formatted in a table or need explanations for any specific one!