Precalculus - Fractions Worksheet - Free Printable

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

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

✔ 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 $1\frac{1}{3} = \frac{4}{3}$

Common denominator: 24

$$
\frac{13}{8} = \frac{39}{24},\quad \frac{4}{3} = \frac{32}{24}
$$

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

✔ Answer: $\frac{7}{24}$

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

Add negatives:

$$
= -\frac{1}{2} - \frac{7}{8} = -\frac{4}{8} - \frac{7}{8} = -\frac{11}{8}
$$

✔ Answer: $-\frac{11}{8}$ or $-1\frac{3}{8}$

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

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

Common denominator: 35

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

✔ Answer: $-\frac{38}{35}$ or $-1\frac{3}{35}$

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

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

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

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

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

$$
= -2 - 1.5 = -3.5 = -\frac{7}{2}
$$

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

✔ Answer: $-\frac{7}{2}$ or $-3\frac{1}{2}$

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

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

So:
$$
-1 - (-\frac{13}{4}) = -1 + \frac{13}{4} = -\frac{4}{4} + \frac{13}{4} = \frac{9}{4}
$$

✔ Answer: $\frac{9}{4}$ or $2\frac{1}{4}$

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

Common denominator: 4

$$
= -\frac{1}{4} + \frac{14}{4} - \frac{3}{4} = \frac{10}{4} = \frac{5}{2}
$$

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

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

Convert all to common denominator: 12

$$
\frac{3}{4} = \frac{9}{12},\quad \frac{7}{6} = \frac{14}{12},\quad 2 = \frac{24}{12}
$$

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

✔ Answer: $-\frac{29}{12}$ or $-2\frac{5}{12}$

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#### 13) $\left(-\frac{3}{4}\right) + \frac{2}{3} + 3 - 1\frac{5}{7}$

Convert all to improper fractions:

- $3 = \frac{3}{1}$
- $1\frac{5}{7} = \frac{12}{7}$

Now:
$$
= -\frac{3}{4} + \frac{2}{3} + \frac{3}{1} - \frac{12}{7}
$$

Find LCD of 4, 3, 1, 7 → LCM(4,3,7) = 84

Convert:

- $-\frac{3}{4} = -\frac{63}{84}$
- $\frac{2}{3} = \frac{56}{84}$
- $\frac{3}{1} = \frac{252}{84}$
- $-\frac{12}{7} = -\frac{144}{84}$

Now add:
$$
-63 + 56 + 252 - 144 = (-63 + 56) = -7,\quad (-7 + 252) = 245,\quad (245 - 144) = 101
$$

So: $\frac{101}{84}$

✔ Answer: $\frac{101}{84}$ or $1\frac{17}{84}$

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

First, compute $-4(-2) = +8$

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

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

Convert to common denominator: 6

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

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

✔ Answer: $\frac{11}{6}$ or $1\frac{5}{6}$

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

Convert all:

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

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

LCD of 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:
$$
30 + 54 + 95 - 105 = 74
$$

So: $\frac{74}{30} = \frac{37}{15}$

✔ Answer: $\frac{37}{15}$ or $2\frac{7}{15}$

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

Simplify:

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

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

Convert:

- $2 = \frac{42}{21}$, better LCD of 3 and 7 is 21

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

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

✔ Answer: $\frac{62}{21}$ or $2\frac{20}{21}$

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Part 2: Simplify Each Expression

Now we simplify algebraic expressions with fractions.

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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:

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

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

Group:

- $\frac{4}{3}x^2$
- $\frac{14}{3}x + \frac{7}{2}x$
- $-\frac{1}{2}$

First, combine $x$ terms:

LCD of 3 and 2 is 6:

- $\frac{14}{3}x = \frac{28}{6}x$
- $\frac{7}{2}x = \frac{21}{6}x$
- Sum: $\frac{49}{6}x$

Constant: $-\frac{1}{2}$

So:
$$
\frac{4}{3}x^2 + \frac{49}{6}x - \frac{1}{2}
$$

✔ Answer: $\frac{4}{3}x^2 + \frac{49}{6}x - \frac{1}{2}$

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

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

Now:

- $2x^3 + \frac{5}{2}x^3 = \left(2 + \frac{5}{2}\right)x^3 = \frac{4}{2} + \frac{5}{2} = \frac{9}{2}x^3$
- Constants: $\frac{2}{3} + \frac{7}{2}$

LCD of 3 and 2 is 6:

- $\frac{2}{3} = \frac{4}{6},\quad \frac{7}{2} = \frac{21}{6} → \frac{25}{6}$

So:
$$
\frac{9}{2}x^3 + \frac{25}{6}
$$

✔ Answer: $\frac{9}{2}x^3 + \frac{25}{6}$

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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:

- $x^3$: $\frac{19}{5} - \frac{2}{3}$

LCD = 15:

- $\frac{19}{5} = \frac{57}{15},\quad \frac{2}{3} = \frac{10}{15} → \frac{47}{15}x^3$

- $x^2$: $2x^2$

- $x$: $\frac{25}{8}x$

No constant.

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:

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

LCD of 7 and 5 = 35

- $-\frac{10}{7} = -\frac{50}{35},\quad \frac{1}{5} = \frac{7}{35} → -\frac{43}{35}x^3$

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

LCD of 3 and 6 = 6

- $-\frac{4}{3} = -\frac{8}{6},\quad \frac{29}{6} → \frac{21}{6} = \frac{7}{2}$

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 | $\frac{101}{84}$ |
| 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 | $\frac{4}{3}x^2 + \frac{49}{6}x - \frac{1}{2}$ |
| 19 | $\frac{9}{2}x^3 + \frac{25}{6}$ |
| 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 | $-\frac{43}{35}x^3 + \frac{7}{2}$ |

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