Operations on Rational Numbers worksheet - Free Printable

Let's solve each problem step by step, following the format provided in the worksheet. We'll perform operations with rational numbers (fractions) and simplify answers.

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1) $\frac{2}{3} + \frac{7}{2}$

To add fractions, find a common denominator. The least common denominator (LCD) of 3 and 2 is 6.

$$
\frac{2}{3} + \frac{7}{2} = \frac{2 \cdot 2}{3 \cdot 2} + \frac{7 \cdot 3}{2 \cdot 3} = \frac{4}{6} + \frac{21}{6} = \frac{25}{6}
$$

Now fill in the blanks:

$$
= \frac{2(2) + 7(3)}{(3)(2)} = \frac{4 + 21}{6} = \frac{25}{6}
$$

✔ Answer: $\boxed{\frac{25}{6}}$

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2) $-\frac{4}{5} + \frac{7}{10}$

LCD of 5 and 10 is 10.

$$
-\frac{4}{5} = -\frac{8}{10}, \quad \frac{7}{10} = \frac{7}{10}
$$
$$
-\frac{8}{10} + \frac{7}{10} = \frac{-1}{10}
$$

Using the given format:

$$
= \frac{-4(2) + 7(1)}{(5)(2)} = \frac{-8 + 7}{10} = \frac{-1}{10}
$$

✔ Answer: $\boxed{-\frac{1}{10}}$

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

Write 5 as $\frac{5}{1}$, then subtract:

$$
\frac{6}{5} - \frac{5}{1} = \frac{6 \cdot 1 - 5 \cdot 5}{5 \cdot 1} = \frac{6 - 25}{5} = \frac{-19}{5}
$$

Fill in:

$$
= \frac{6(1) - 5(5)}{(5)(1)} = \frac{6 - 25}{5} = \frac{-19}{5}
$$

✔ Answer: $\boxed{-\frac{19}{5}}$

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

Convert to common denominator: LCD of 4 and 2 is 4

$$
-\frac{7}{4} - \frac{6}{4} = \frac{-7 - 6}{4} = \frac{-13}{4}
$$

Using format:

$$
= \frac{(-7)(2) - (3)(4)}{(4)(2)} = \frac{-14 - 12}{8} = \frac{-26}{8}
$$

Wait! That gives $\frac{-26}{8}$, but we can simplify it:

$$
\frac{-26}{8} = \frac{-13}{4}
$$

But let’s check: Is this correct?

Actually, better to use LCM = 4, so:

$$
-\frac{7}{4} - \frac{3}{2} = -\frac{7}{4} - \frac{6}{4} = \frac{-13}{4}
$$

So the format should be:

$$
= \frac{(-7)(1) - (3)(2)}{(4)(1)} = \frac{-7 - 6}{4} = \frac{-13}{4}
$$

But the blank format has two denominators: $(\square)(\square)$, so likely they want cross-multiplication method:

$$
= \frac{(-7)(2) - (3)(4)}{(4)(2)} = \frac{-14 - 12}{8} = \frac{-26}{8} = \frac{-13}{4}
$$

Yes — simplifying $\frac{-26}{8}$ → divide numerator and denominator by 2: $\frac{-13}{4}$

So:

$$
= \frac{(-7)(2) - (3)(4)}{(4)(2)} = \frac{-14 - 12}{8} = \frac{-26}{8} = \frac{-13}{4}
$$

✔ Answer: $\boxed{-\frac{13}{4}}$

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5) $\frac{2}{3} \cdot \left(-\frac{7}{4}\right)$

Multiply numerators and denominators:

$$
= \frac{2 \cdot (-7)}{3 \cdot 4} = \frac{-14}{12} = \frac{-7}{6}
$$

Simplify: divide numerator and denominator by 2.

$$
= \frac{-7}{6}
$$

So:

$$
= \frac{-14}{12} = \frac{-7}{6}
$$

✔ Answer: $\boxed{-\frac{7}{6}}$

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6) $-4 \cdot \left(-\frac{3}{2}\right)$

First, write $-4$ as $-\frac{4}{1}$

$$
= \frac{-4}{1} \cdot \frac{-3}{2} = \frac{(-4)(-3)}{1 \cdot 2} = \frac{12}{2} = 6
$$

So:

$$
= \frac{12}{2} = 6
$$

✔ Answer: $\boxed{6}$

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7) $-\frac{3}{4} \div \frac{2}{5}$

Dividing fractions: multiply by reciprocal.

$$
= -\frac{3}{4} \cdot \frac{5}{2} = \frac{(-3)(5)}{4 \cdot 2} = \frac{-15}{8}
$$

✔ Answer: $\boxed{-\frac{15}{8}}$

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✔ Final Answers Summary:

1) $\frac{25}{6}$
2) $-\frac{1}{10}$
3) $-\frac{19}{5}$
4) $-\frac{13}{4}$
5) $-\frac{7}{6}$
6) $6$
7) $-\frac{15}{8}$

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🔍 Explanation of Key Concepts:

- Addition/Subtraction: Find common denominator, then combine numerators.
- Multiplication: Multiply numerators and denominators directly.
- Division: Multiply by the reciprocal.
- Always simplify the final answer (reduce fraction if possible).
- Signs matter: negative × negative = positive; negative × positive = negative.

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