PreCalculus worksheet featuring fraction operations and algebraic simplification exercises.

Let's solve each problem step by step from the Precalculus Fractions Worksheet. We'll evaluate and simplify expressions involving fractions, integers, and algebraic terms.

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Part 1: Evaluate Each Expression

We will evaluate each expression using fraction arithmetic rules (common denominators, signs, etc.).

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#### 1) $\left(-\frac{1}{6}\right) - \frac{2}{5}$

Find a common denominator: $30$

$$
= \left(-\frac{5}{30}\right) - \frac{12}{30} = -\frac{17}{30}
$$

✔ Answer: $-\frac{17}{30}$

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

Convert to improper fractions:

- $-2\frac{1}{2} = -\frac{5}{2}$
- $2\frac{2}{3} = \frac{8}{3}$

Common denominator: 6

$$
= -\frac{15}{6} + \frac{16}{6} = \frac{1}{6}
$$

✔ Answer: $\frac{1}{6}$

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

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

So:
$$
= -\frac{1}{4} - \frac{4}{3}
$$

Common denominator: 12

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

Or as mixed number: $-1\frac{7}{12}$

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

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

Subtracting a negative is adding:

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

Common denominator: 8

$$
= -\frac{2}{8} + \frac{3}{8} = \frac{1}{8}
$$

✔ Answer: $\frac{1}{8}$

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#### 5) $\frac{13}{8} - 1\frac{1}{3}$

Convert $1\frac{1}{3} = \frac{4}{3}$

Now:
$$
= \frac{13}{8} - \frac{4}{3}
$$

LCM of 8 and 3 is 24:

$$
= \frac{39}{24} - \frac{32}{24} = \frac{7}{24}
$$

✔ Answer: $\frac{7}{24}$

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#### 6) $\left(-\frac{1}{2}\right) + \left(-\frac{7}{8}\right)$

Same sign → add absolute values, keep negative:

$$
= -\left(\frac{1}{2} + \frac{7}{8}\right) = -\left(\frac{4}{8} + \frac{7}{8}\right) = -\frac{11}{8}
$$

Or: $-1\frac{3}{8}$

✔ Answer: $-\frac{11}{8}$

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#### 7) $\frac{5}{7} + \left(-\frac{9}{5}\right)$

$$
= \frac{5}{7} - \frac{9}{5}
$$

LCM of 7 and 5 is 35:

$$
= \frac{25}{35} - \frac{63}{35} = -\frac{38}{35}
$$

Or: $-1\frac{3}{35}$

✔ Answer: $-\frac{38}{35}$

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#### 8) $(-2) + 3\frac{1}{2}$

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

$$
= -2 + \frac{7}{2} = -\frac{4}{2} + \frac{7}{2} = \frac{3}{2}
$$

Or: $1\frac{1}{2}$

✔ Answer: $\frac{3}{2}$

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#### 9) $(-2) - \frac{3}{2}$

$$
= -2 - \frac{3}{2} = -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2}
$$

Or: $-3\frac{1}{2}$

✔ Answer: $-\frac{7}{2}$

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#### 10) $(-1) - \left(-3\frac{1}{4}\right)$

Subtracting a negative → add:

$$
= -1 + 3\frac{1}{4} = -1 + \frac{13}{4} = -\frac{4}{4} + \frac{13}{4} = \frac{9}{4}
$$

Or: $2\frac{1}{4}$

✔ Answer: $\frac{9}{4}$

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#### 11) $\left(-\frac{2}{8}\right) + 3\frac{1}{2} - \frac{3}{4}$

Simplify $-\frac{2}{8} = -\frac{1}{4}$

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

Now:
$$
= -\frac{1}{4} + \frac{7}{2} - \frac{3}{4}
$$

Convert all to fourths:
$$
= -\frac{1}{4} + \frac{14}{4} - \frac{3}{4} = \frac{10}{4} = \frac{5}{2}
$$

Or: $2\frac{1}{2}$

✔ Answer: $\frac{5}{2}$

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#### 12) $\frac{3}{4} - \frac{7}{6} - 2$

Convert to common denominator (LCM of 4, 6, 1 is 12):

$$
= \frac{9}{12} - \frac{14}{12} - \frac{24}{12} = \frac{9 - 14 - 24}{12} = \frac{-29}{12}
$$

Or: $-2\frac{5}{12}$

✔ Answer: $-\frac{29}{12}$

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#### 13) $\left(-\frac{3}{4}\right) + \frac{2}{3} + 3 - 1\frac{6}{7}$

Convert all to improper fractions where needed:

- $3 = \frac{3}{1}$
- $1\frac{6}{7} = \frac{13}{7}$

So:
$$
= -\frac{3}{4} + \frac{2}{3} + \frac{3}{1} - \frac{13}{7}
$$

Find LCM of 4, 3, 1, 7 → 84

Convert each:

- $-\frac{3}{4} = -\frac{63}{84}$
- $\frac{2}{3} = \frac{56}{84}$
- $\frac{3}{1} = \frac{252}{84}$
- $-\frac{13}{7} = -\frac{156}{84}$

Add:
$$
= -63 + 56 + 252 - 156 = (-63 + 56) = -7; (252 - 156) = 96 → -7 + 96 = 89
$$

So: $\frac{89}{84}$

Or: $1\frac{5}{84}$

✔ Answer: $\frac{89}{84}$

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

First: $-4(-2) = +8$

$4\frac{2}{3} = \frac{14}{3}$

So:
$$
= -\frac{3}{2} + 8 - \frac{14}{3}
$$

Convert to common denominator (LCM of 2,1,3 is 6)

- $-\frac{3}{2} = -\frac{9}{6}$
- $8 = \frac{48}{6}$
- $-\frac{14}{3} = -\frac{28}{6}$

Now:
$$
= -\frac{9}{6} + \frac{48}{6} - \frac{28}{6} = \frac{11}{6}
$$

Or: $1\frac{5}{6}$

✔ Answer: $\frac{11}{6}$

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#### 15) $1 + \frac{9}{5} + 3\frac{1}{6} + \left(-3\frac{1}{2}\right)$

Convert all:

- $1 = 1$
- $\frac{9}{5} = \frac{9}{5}$
- $3\frac{1}{6} = \frac{19}{6}$
- $-3\frac{1}{2} = -\frac{7}{2}$

Now:
$$
= 1 + \frac{9}{5} + \frac{19}{6} - \frac{7}{2}
$$

LCM of 1,5,6,2 → 30

Convert:

- $1 = \frac{30}{30}$
- $\frac{9}{5} = \frac{54}{30}$
- $\frac{19}{6} = \frac{95}{30}$
- $-\frac{7}{2} = -\frac{105}{30}$

Add:
$$
= \frac{30 + 54 + 95 - 105}{30} = \frac{74}{30} = \frac{37}{15}
$$

Or: $2\frac{7}{15}$

✔ Answer: $\frac{37}{15}$

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#### 16) $\left(-\frac{2}{3}\right) - \left(-\frac{1}{3}\right) + 2 - \left(-\frac{9}{7}\right)$

Simplify negatives:

- $-\left(-\frac{1}{3}\right) = +\frac{1}{3}$
- $-\left(-\frac{9}{7}\right) = +\frac{9}{7}$

So:
$$
= -\frac{2}{3} + \frac{1}{3} + 2 + \frac{9}{7}
$$

Combine like terms:

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

Now:
$$
= -\frac{1}{3} + 2 + \frac{9}{7}
$$

Convert to common denominator (LCM of 3,1,7 = 21)

- $-\frac{1}{3} = -\frac{7}{21}$
- $2 = \frac{42}{21}$
- $\frac{9}{7} = \frac{27}{21}$

Add:
$$
= -\frac{7}{21} + \frac{42}{21} + \frac{27}{21} = \frac{62}{21}
$$

Or: $2\frac{20}{21}$

✔ Answer: $\frac{62}{21}$

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Part 2: Simplify Each Expression

These are algebraic expressions with variables. Combine like terms.

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#### 17) $\left(-\frac{3}{2}x - 1\right) + \left(-\frac{5}{3}x^2 + \frac{4}{3}\right)$

Group like terms:

- $-\frac{5}{3}x^2$
- $-\frac{3}{2}x$
- $-1 + \frac{4}{3} = \frac{-3 + 4}{3} = \frac{1}{3}$

So:
$$
= -\frac{5}{3}x^2 - \frac{3}{2}x + \frac{1}{3}
$$

✔ Answer: $-\frac{5}{3}x^2 - \frac{3}{2}x + \frac{1}{3}$

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#### 18) $\left(\frac{14}{3} - \frac{1}{2}\right) + \left(\frac{7}{2}x + \frac{4}{3}x^2\right)$

First: $\frac{14}{3} - \frac{1}{2}$

LCM = 6:

$$
= \frac{28}{6} - \frac{3}{6} = \frac{25}{6}
$$

Then: $\frac{7}{2}x + \frac{4}{3}x^2$

So combine:

$$
= \frac{4}{3}x^2 + \frac{7}{2}x + \frac{25}{6}
$$

✔ Answer: $\frac{4}{3}x^2 + \frac{7}{2}x + \frac{25}{6}$

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#### 19) $\left(\frac{12}{6}x^3 + \frac{2}{3}\right) + \left(\frac{7}{2} + \frac{5}{2}x^4\right)$

Simplify: $\frac{12}{6} = 2$, so $2x^3$

Now group:

- $\frac{5}{2}x^4$
- $2x^3$
- $\frac{2}{3} + \frac{7}{2} = \frac{4}{6} + \frac{21}{6} = \frac{25}{6}$

So:
$$
= \frac{5}{2}x^4 + 2x^3 + \frac{25}{6}
$$

✔ Answer: $\frac{5}{2}x^4 + 2x^3 + \frac{25}{6}$

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#### 20) $\left(\frac{19}{5}x^3 + 2x^2\right) - \left(\frac{2}{3}x^3 - \frac{25}{8}x\right)$

Distribute the minus sign:

$$
= \frac{19}{5}x^3 + 2x^2 - \frac{2}{3}x^3 + \frac{25}{8}x
$$

Combine like terms:

- $x^3$: $\frac{19}{5} - \frac{2}{3} = \frac{57 - 10}{15} = \frac{47}{15}$
- $x^2$: $2x^2$
- $x$: $\frac{25}{8}x$

So:
$$
= \frac{47}{15}x^3 + 2x^2 + \frac{25}{8}x
$$

✔ Answer: $\frac{47}{15}x^3 + 2x^2 + \frac{25}{8}x$

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#### 21) $\left(-\frac{5}{3}x + \frac{5}{4}x^2\right) + \left(\frac{1}{4}x^2 - 2x^3\right)$

Group:

- $x^3$: $-2x^3$
- $x^2$: $\frac{5}{4}x^2 + \frac{1}{4}x^2 = \frac{6}{4}x^2 = \frac{3}{2}x^2$
- $x$: $-\frac{5}{3}x$

So:
$$
= -2x^3 + \frac{3}{2}x^2 - \frac{5}{3}x
$$

✔ Answer: $-2x^3 + \frac{3}{2}x^2 - \frac{5}{3}x$

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#### 22) $\left(-\frac{10}{7}x^3 - \frac{4}{3}\right) + \left(\frac{29}{6} + \frac{1}{5}x^3\right)$

Group:

- $x^3$: $-\frac{10}{7}x^3 + \frac{1}{5}x^3$

LCM of 7 and 5 is 35:

$$
= -\frac{50}{35}x^3 + \frac{7}{35}x^3 = -\frac{43}{35}x^3
$$

Constants: $-\frac{4}{3} + \frac{29}{6}$

LCM = 6:

$$
= -\frac{8}{6} + \frac{29}{6} = \frac{21}{6} = \frac{7}{2}
$$

So:
$$
= -\frac{43}{35}x^3 + \frac{7}{2}
$$

✔ Answer: $-\frac{43}{35}x^3 + \frac{7}{2}$

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✔ Final Answers Summary

| Problem | Answer |
|--------|--------|
| 1 | $-\frac{17}{30}$ |
| 2 | $\frac{1}{6}$ |
| 3 | $-\frac{19}{12}$ |
| 4 | $\frac{1}{8}$ |
| 5 | $\frac{7}{24}$ |
| 6 | $-\frac{11}{8}$ |
| 7 | $-\frac{38}{35}$ |
| 8 | $\frac{3}{2}$ |
| 9 | $-\frac{7}{2}$ |
| 10 | $\frac{9}{4}$ |
| 11 | $\frac{5}{2}$ |
| 12 | $-\frac{29}{12}$ |
| 13 | $\frac{89}{84}$ |
| 14 | $\frac{11}{6}$ |
| 15 | $\frac{37}{15}$ |
| 16 | $\frac{62}{21}$ |
| 17 | $-\frac{5}{3}x^2 - \frac{3}{2}x + \frac{1}{3}$ |
| 18 | $\frac{4}{3}x^2 + \frac{7}{2}x + \frac{25}{6}$ |
| 19 | $\frac{5}{2}x^4 + 2x^3 + \frac{25}{6}$ |
| 20 | $\frac{47}{15}x^3 + 2x^2 + \frac{25}{8}x$ |
| 21 | $-2x^3 + \frac{3}{2}x^2 - \frac{5}{3}x$ |
| 22 | $-\frac{43}{35}x^3 + \frac{7}{2}$ |

Let me know if you'd like these answers boxed or formatted differently!