Fraction Arithmetic 7A664C

1. **Problem:** Calculate $\frac{4}{5} \cdot \frac{25}{12}$. **Step 1:** Multiply numerators and denominators: $$\frac{4 \times 25}{5 \times 12} = \frac{100}{60}$$ **Step 2:** Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD), which is 20: $$\frac{\cancel{100}^{5 \times 20}}{\cancel{60}^{3 \times 20}} = \frac{5}{3}$$ **Answer:** $\frac{5}{3}$. 2. **Problem:** Calculate $\left|\frac{5}{3} - \frac{5}{6}\right| : 3$. **Step 1:** Find the difference inside the absolute value: $$\frac{5}{3} - \frac{5}{6} = \frac{10}{6} - \frac{5}{6} = \frac{5}{6}$$ **Step 2:** Absolute value of $\frac{5}{6}$ is $\frac{5}{6}$. **Step 3:** Divide by 3: $$\frac{5}{6} : 3 = \frac{5}{6} \times \frac{1}{3} = \frac{5}{18}$$ **Answer:** $\frac{5}{18}$. 3. **Problem:** Calculate $\left|\frac{3}{4} : \frac{25}{6}\right|$. **Step 1:** Division of fractions means multiplying by the reciprocal: $$\frac{3}{4} : \frac{25}{6} = \frac{3}{4} \times \frac{6}{25} = \frac{18}{100}$$ **Step 2:** Simplify $\frac{18}{100}$ by dividing numerator and denominator by 2: $$\frac{\cancel{18}^{9 \times 2}}{\cancel{100}^{50 \times 2}} = \frac{9}{50}$$ **Step 3:** Absolute value of $\frac{9}{50}$ is $\frac{9}{50}$. **Answer:** $\frac{9}{50}$. 4. **Problem:** Calculate $\left(\frac{3}{2} - \frac{2}{3}\right) : \left(1 - \frac{5}{4}\right)$. **Step 1:** Calculate numerator: $$\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}$$ **Step 2:** Calculate denominator: $$1 - \frac{5}{4} = \frac{4}{4} - \frac{5}{4} = -\frac{1}{4}$$ **Step 3:** Divide numerator by denominator: $$\frac{5}{6} : \left(-\frac{1}{4}\right) = \frac{5}{6} \times \left(-4\right) = -\frac{20}{6}$$ **Step 4:** Simplify $-\frac{20}{6}$ by dividing numerator and denominator by 2: $$-\frac{\cancel{20}^{10 \times 2}}{\cancel{6}^{3 \times 2}} = -\frac{10}{3}$$ **Answer:** $-\frac{10}{3}$. 5. **Problem:** Calculate $\frac{\frac{2}{3}}{\frac{4}{5}}$. **Step 1:** Division of fractions means multiplying by the reciprocal: $$\frac{2}{3} : \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12}$$ **Step 2:** Simplify $\frac{10}{12}$ by dividing numerator and denominator by 2: $$\frac{\cancel{10}^{5 \times 2}}{\cancel{12}^{6 \times 2}} = \frac{5}{6}$$ **Answer:** $\frac{5}{6}$. 6. **Problem:** Calculate $\frac{\frac{3}{2} - \frac{2}{3}}{\frac{4}{2} - \frac{3}{4}}$. **Step 1:** Calculate numerator: $$\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}$$ **Step 2:** Calculate denominator: $$\frac{4}{2} - \frac{3}{4} = 2 - \frac{3}{4} = \frac{8}{4} - \frac{3}{4} = \frac{5}{4}$$ **Step 3:** Divide numerator by denominator: $$\frac{\frac{5}{6}}{\frac{5}{4}} = \frac{5}{6} \times \frac{4}{5} = \frac{20}{30}$$ **Step 4:** Simplify $\frac{20}{30}$ by dividing numerator and denominator by 10: $$\frac{\cancel{20}^{2 \times 10}}{\cancel{30}^{3 \times 10}} = \frac{2}{3}$$ **Answer:** $\frac{2}{3}$.