Operations on Rational Numbers worksheet - Free Printable

Problem: Operations in Rational Numbers

We are tasked with performing the given operations on rational numbers and simplifying the answers. Let's solve each problem step by step.

---

#### 1) \( \frac{2}{3} + \frac{7}{2} \)

To add fractions, we need a common denominator. The denominators are 3 and 2. The least common denominator (LCD) is 6.

- Rewrite each fraction with the denominator 6:
\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\]
\[
\frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6}
\]

- Add the fractions:
\[
\frac{2}{3} + \frac{7}{2} = \frac{4}{6} + \frac{21}{6} = \frac{4 + 21}{6} = \frac{25}{6}
\]

- Simplify (if possible):
\[
\frac{25}{6} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{\frac{25}{6}}
\]

---

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

The denominators are 5 and 10. The LCD is 10.

- Rewrite each fraction with the denominator 10:
\[
-\frac{4}{5} = -\frac{4 \times 2}{5 \times 2} = -\frac{8}{10}
\]
\[
\frac{7}{10} \text{ remains } \frac{7}{10}
\]

- Add the fractions:
\[
-\frac{4}{5} + \frac{7}{10} = -\frac{8}{10} + \frac{7}{10} = \frac{-8 + 7}{10} = \frac{-1}{10}
\]

- Simplify (if possible):
\[
\frac{-1}{10} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{-\frac{1}{10}}
\]

---

#### 3) \( \frac{6}{5} - 5 \)

Rewrite 5 as a fraction with the same denominator as \( \frac{6}{5} \):
\[
5 = \frac{5 \times 5}{1 \times 5} = \frac{25}{5}
\]

- Subtract the fractions:
\[
\frac{6}{5} - 5 = \frac{6}{5} - \frac{25}{5} = \frac{6 - 25}{5} = \frac{-19}{5}
\]

- Simplify (if possible):
\[
\frac{-19}{5} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{-\frac{19}{5}}
\]

---

#### 4) \( -\frac{7}{4} - \frac{3}{2} \)

The denominators are 4 and 2. The LCD is 4.

- Rewrite each fraction with the denominator 4:
\[
-\frac{7}{4} \text{ remains } -\frac{7}{4}
\]
\[
\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}
\]

- Subtract the fractions:
\[
-\frac{7}{4} - \frac{3}{2} = -\frac{7}{4} - \frac{6}{4} = \frac{-7 - 6}{4} = \frac{-13}{4}
\]

- Simplify (if possible):
\[
\frac{-13}{4} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{-\frac{13}{4}}
\]

---

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

Multiply the numerators and the denominators:
\[
\frac{2}{3} \cdot \left( -\frac{7}{4} \right) = \frac{2 \cdot (-7)}{3 \cdot 4} = \frac{-14}{12}
\]

- Simplify the fraction:
\[
\frac{-14}{12} = \frac{-7}{6} \quad \text{(divide numerator and denominator by 2)}
\]

Answer:
\[
\boxed{-\frac{7}{6}}
\]

---

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

Multiply the integers and the fraction:
\[
-4 \cdot \left( -\frac{3}{2} \right) = \frac{-4 \cdot (-3)}{2} = \frac{12}{2} = 6
\]

Answer:
\[
\boxed{6}
\]

---

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

Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[
-\frac{3}{4} \div \frac{2}{5} = -\frac{3}{4} \cdot \frac{5}{2} = \frac{-3 \cdot 5}{4 \cdot 2} = \frac{-15}{8}
\]

- Simplify (if possible):
\[
\frac{-15}{8} \text{ is already in simplest form.}
\]

Answer:
\[
\boxed{-\frac{15}{8}}
\]

---

Final Answers:

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