Sums & Products of Rational Numbers Quick Check

Sure! To check your understanding of the sums and products of rational numbers, here are some quick questions along with their solutions:

### Quick Check Questions

1. **Sum of Rational Numbers**:
What is the sum of \( \frac{3}{4} \) and \( \frac{2}{3} \)?

2. **Product of Rational Numbers**:
What is the product of \( \frac{5}{6} \) and \( \frac{4}{9} \)?

3. **Sum with Mixed Numbers**:
What is the sum of \( 2 \frac{1}{2} \) and \( 1 \frac{3}{4} \)?

4. **Product of a Rational Number and an Integer**:
What is the product of \( \frac{7}{8} \) and \( 3 \)?

5. **Sum of Negative Rational Numbers**:
What is the sum of \( -\frac{2}{5} \) and \( -\frac{1}{3} \)?

### Solutions

1. **Sum of \( \frac{3}{4} \) and \( \frac{2}{3} \)**:
\[
\frac{3}{4} + \frac{2}{3} = \frac{3 \cdot 3}{4 \cdot 3} + \frac{2 \cdot 4}{3 \cdot 4} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12}
\]

2. **Product of \( \frac{5}{6} \) and \( \frac{4}{9} \)**:
\[
\frac{5}{6} \times \frac{4}{9} = \frac{5 \cdot 4}{6 \cdot 9} = \frac{20}{54} = \frac{10}{27} \quad (\text{after simplification})
\]

3. **Sum of \( 2 \frac{1}{2} \) and \( 1 \frac{3}{4} \)**:
\[
2 \frac{1}{2} = \frac{5}{2}, \quad 1 \frac{3}{4} = \frac{7}{4}
\]
\[
\frac{5}{2} + \frac{7}{4} = \frac{5 \cdot 2}{2 \cdot 2} + \frac{7 \cdot 1}{4 \cdot 1} = \frac{10}{4} + \frac{7}{4} = \frac{17}{4}
\]

4. **Product of \( \frac{7}{8} \) and \( 3 \)**:
\[
\frac{7}{8} \times 3 = \frac{7 \cdot 3}{8} = \frac{21}{8}
\]

5. **Sum of \( -\frac{2}{5} \) and \( -\frac{1}{3} \)**:
\[
-\frac{2}{5} + -\frac{1}{3} = -\left(\frac{2}{5} + \frac{1}{3}\right) = -\left(\frac{2 \cdot 3}{5 \cdot 3} + \frac{1 \cdot 5}{3 \cdot 5}\right) = -\left(\frac{6}{15} + \frac{5}{15}\right) = -\frac{11}{15}
\]

Feel free to ask more questions if you'd like further practice or explanation!