Arithmetic Operations 508B4D

1. **Problem:** Calculate $92.516 - 12.531 + 41.535$. **Step 1:** Perform subtraction first: $$92.516 - 12.531 = 79.985$$ **Step 2:** Add the result to $41.535$: $$79.985 + 41.535 = 121.52$$ **Answer:** $121.52$ 2. **Problem:** Multiply fractions $\frac{5}{16} \cdot \frac{2}{3}$. **Step 1:** Multiply numerators and denominators: $$\frac{5 \times 2}{16 \times 3} = \frac{10}{48}$$ **Step 2:** Simplify fraction by dividing numerator and denominator by 2: $$\frac{\cancel{10}^5}{\cancel{48}^{24}} = \frac{5}{24}$$ **Answer:** $\frac{5}{24}$ 3. **Problem:** Add mixed numbers $4 \frac{6}{8} + 1 \frac{5}{6}$. **Step 1:** Convert mixed numbers to improper fractions: $$4 \frac{6}{8} = 4 + \frac{6}{8} = \frac{32}{8} + \frac{6}{8} = \frac{38}{8}$$ $$1 \frac{5}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6}$$ **Step 2:** Find common denominator for $8$ and $6$, which is $24$. **Step 3:** Convert fractions: $$\frac{38}{8} = \frac{38 \times 3}{8 \times 3} = \frac{114}{24}$$ $$\frac{11}{6} = \frac{11 \times 4}{6 \times 4} = \frac{44}{24}$$ **Step 4:** Add fractions: $$\frac{114}{24} + \frac{44}{24} = \frac{158}{24}$$ **Step 5:** Simplify fraction by dividing numerator and denominator by 2: $$\frac{\cancel{158}^{79}}{\cancel{24}^{12}} = \frac{79}{12}$$ **Step 6:** Convert back to mixed number: $$\frac{79}{12} = 6 \frac{7}{12}$$ **Answer:** $6 \frac{7}{12}$ 4. **Problem:** Calculate $1 - \frac{4}{5} + \frac{1}{2}$. **Step 1:** Find common denominator for $5$ and $2$, which is $10$. **Step 2:** Convert fractions: $$1 = \frac{10}{10}$$ $$\frac{4}{5} = \frac{8}{10}$$ $$\frac{1}{2} = \frac{5}{10}$$ **Step 3:** Perform operations: $$\frac{10}{10} - \frac{8}{10} + \frac{5}{10} = \frac{10 - 8 + 5}{10} = \frac{7}{10}$$ **Answer:** $\frac{7}{10}$ 5. **Problem:** Multiply decimals $(4.24)(3.14)$. **Step 1:** Multiply as whole numbers ignoring decimals: $$424 \times 314 = 133136$$ **Step 2:** Count decimal places: 2 in $4.24$ and 2 in $3.14$, total 4. **Step 3:** Place decimal point 4 places from right: $$13.3136$$ **Answer:** $13.3136$ 6. **Problem:** Divide decimals $\frac{6.89}{1.56}$. **Step 1:** Convert division to fraction: $$\frac{6.89}{1.56}$$ **Step 2:** Multiply numerator and denominator by 100 to clear decimals: $$\frac{689}{156}$$ **Step 3:** Perform division: $$689 \div 156 \approx 4.4179$$ **Answer:** Approximately $4.4179$ 7. **Problem:** Calculate $17.425 - 12.512 - 1.435$. **Step 1:** Subtract first two numbers: $$17.425 - 12.512 = 4.913$$ **Step 2:** Subtract the last number: $$4.913 - 1.435 = 3.478$$ **Answer:** $3.478$ 8. **Problem:** Multiply mixed numbers $4 \frac{5}{2} \cdot 6 \frac{1}{3}$. **Step 1:** Convert mixed numbers to improper fractions: $$4 \frac{5}{2} = 4 + \frac{5}{2} = \frac{8}{2} + \frac{5}{2} = \frac{13}{2}$$ $$6 \frac{1}{3} = 6 + \frac{1}{3} = \frac{18}{3} + \frac{1}{3} = \frac{19}{3}$$ **Step 2:** Multiply fractions: $$\frac{13}{2} \times \frac{19}{3} = \frac{247}{6}$$ **Step 3:** Convert to mixed number: $$\frac{247}{6} = 41 \frac{1}{6}$$ **Answer:** $41 \frac{1}{6}$ 9. **Problem:** Divide decimals $\frac{42.12}{20}$. **Step 1:** Perform division: $$42.12 \div 20 = 2.106$$ **Answer:** $2.106$ 10. **Problem:** Calculate $\frac{72}{150 \div 2}$. **Step 1:** Calculate denominator division: $$150 \div 2 = 75$$ **Step 2:** Divide numerator by denominator: $$\frac{72}{75}$$ **Step 3:** Simplify fraction by dividing numerator and denominator by 3: $$\frac{\cancel{72}^{24}}{\cancel{75}^{25}} = \frac{24}{25}$$ **Answer:** $\frac{24}{25}$ **Right side:** 3) Calculate $5 \div \frac{1}{2} \cdot \frac{5}{1}$. **Step 1:** Division by fraction is multiplication by reciprocal: $$5 \div \frac{1}{2} = 5 \times 2 = 10$$ **Step 2:** Multiply by $\frac{5}{1}$: $$10 \times 5 = 50$$ **Answer:** $50$ .5 \div \frac{1}{2}$ **Step 1:** Multiply by reciprocal: $$0.5 \times 2 = 1$$ **Answer:** $1$ .5 - .5 **Step 1:** Subtract: $$0.5 - 0.5 = 0$$ **Answer:** $0$ $\frac{5}{2} \times 2$ **Step 1:** Multiply: $$\frac{5}{2} \times 2 = \frac{5 \times 2}{2} = \frac{\cancel{5 \times \cancel{2}}^{5 \times 1}}{\cancel{2}} = 5$$ **Answer:** $5$ $Y = 50$