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Sagot :
Certainly! Let's solve each problem step by step.
### Problem 1: [tex]\(4 \frac{1}{2} + 6 \frac{1}{5}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(4 \frac{1}{2} = \frac{4 \cdot 2 + 1}{2} = \frac{9}{2}\)[/tex]
- [tex]\(6 \frac{1}{5} = \frac{6 \cdot 5 + 1}{5} = \frac{31}{5}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 2 and 5 is 10.
- Convert [tex]\( \frac{9}{2} \)[/tex] to have a denominator of 10: [tex]\( \frac{9 \cdot 5}{2 \cdot 5} = \frac{45}{10} \)[/tex]
- Convert [tex]\( \frac{31}{5} \)[/tex] to have a denominator of 10: [tex]\( \frac{31 \cdot 2}{5 \cdot 2} = \frac{62}{10} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{45}{10} + \frac{62}{10} = \frac{107}{10} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 107 by 10 to get the quotient and remainder: [tex]\(107 \div 10 = 10\)[/tex] remainder [tex]\(7\)[/tex]
- So, [tex]\( \frac{107}{10} = 10 \frac{7}{10}\)[/tex]
Final result: [tex]\( 4 \frac{1}{2} + 6 \frac{1}{5} = 10 \frac{7}{10} \)[/tex]
### Problem 2: [tex]\(2 \frac{5}{10} + 5 \frac{3}{5}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(2 \frac{5}{10} = \frac{2 \cdot 10 + 5}{10} = \frac{25}{10}\)[/tex]
- [tex]\(5 \frac{3}{5} = \frac{5 \cdot 5 + 3}{5} = \frac{28}{5}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 10 and 5 is 50.
- Convert [tex]\( \frac{25}{10} \)[/tex] to have a denominator of 50: [tex]\( \frac{25 \cdot 5}{10 \cdot 5} = \frac{125}{50} \)[/tex]
- Convert [tex]\( \frac{28}{5} \)[/tex] to have a denominator of 50: [tex]\( \frac{28 \cdot 10}{5 \cdot 10} = \frac{280}{50} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{125}{50} + \frac{280}{50} = \frac{405}{50} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 405 by 50 to get the quotient and remainder: [tex]\(405 \div 50 = 8\)[/tex] remainder [tex]\(5\)[/tex]
- So, [tex]\( \frac{405}{50} = 8 \frac{5}{50}\)[/tex]
Final result: [tex]\( 2 \frac{5}{10} + 5 \frac{3}{5} = 8 \frac{5}{50} \)[/tex]
### Problem 3: [tex]\(3 \frac{4}{5} + 8 \frac{1}{3}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(3 \frac{4}{5} = \frac{3 \cdot 5 + 4}{5} = \frac{19}{5}\)[/tex]
- [tex]\(8 \frac{1}{3} = \frac{8 \cdot 3 + 1}{3} = \frac{25}{3}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 5 and 3 is 15.
- Convert [tex]\( \frac{19}{5} \)[/tex] to have a denominator of 15: [tex]\( \frac{19 \cdot 3}{5 \cdot 3} = \frac{57}{15} \)[/tex]
- Convert [tex]\( \frac{25}{3} \)[/tex] to have a denominator of 15: [tex]\( \frac{25 \cdot 5}{3 \cdot 5} = \frac{125}{15} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{57}{15} + \frac{125}{15} = \frac{182}{15} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 182 by 15 to get the quotient and remainder: [tex]\(182 \div 15 = 12\)[/tex] remainder [tex]\(2\)[/tex]
- So, [tex]\( \frac{182}{15} = 12 \frac{2}{15}\)[/tex]
Final result: [tex]\( 3 \frac{4}{5} + 8 \frac{1}{3} = 12 \frac{2}{15} \)[/tex]
### Problem 4: [tex]\(2 \frac{4}{5} + 8 \frac{3}{4}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(2 \frac{4}{5} = \frac{2 \cdot 5 + 4}{5} = \frac{14}{5}\)[/tex]
- [tex]\(8 \frac{3}{4} = \frac{8 \cdot 4 + 3}{4} = \frac{35}{4}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 5 and 4 is 20.
- Convert [tex]\( \frac{14}{5} \)[/tex] to have a denominator of 20: [tex]\( \frac{14 \cdot 4}{5 \cdot 4} = \frac{56}{20} \)[/tex]
- Convert [tex]\( \frac{35}{4} \)[/tex] to have a denominator of 20: [tex]\( \frac{35 \cdot 5}{4 \cdot 5} = \frac{175}{20} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{56}{20} + \frac{175}{20} = \frac{231}{20} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 231 by 20 to get the quotient and remainder: [tex]\(231 \div 20 = 11\)[/tex] remainder [tex]\(11\)[/tex]
- So, [tex]\( \frac{231}{20} = 11 \frac{11}{20}\)[/tex]
Final result: [tex]\( 2 \frac{4}{5} + 8 \frac{3}{4} = 11 \frac{11}{20} \)[/tex]
Thus, the solved problems are:
1. [tex]\( 4 \frac{1}{2} + 6 \frac{1}{5} = 10 \frac{7}{10} \)[/tex]
2. [tex]\( 2 \frac{5}{10} + 5 \frac{3}{5} = 8 \frac{5}{50} \)[/tex]
3. [tex]\( 3 \frac{4}{5} + 8 \frac{1}{3} = 12 \frac{2}{15} \)[/tex]
4. [tex]\( 2 \frac{4}{5} + 8 \frac{3}{4} = 11 \frac{11}{20} \)[/tex]
### Problem 1: [tex]\(4 \frac{1}{2} + 6 \frac{1}{5}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(4 \frac{1}{2} = \frac{4 \cdot 2 + 1}{2} = \frac{9}{2}\)[/tex]
- [tex]\(6 \frac{1}{5} = \frac{6 \cdot 5 + 1}{5} = \frac{31}{5}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 2 and 5 is 10.
- Convert [tex]\( \frac{9}{2} \)[/tex] to have a denominator of 10: [tex]\( \frac{9 \cdot 5}{2 \cdot 5} = \frac{45}{10} \)[/tex]
- Convert [tex]\( \frac{31}{5} \)[/tex] to have a denominator of 10: [tex]\( \frac{31 \cdot 2}{5 \cdot 2} = \frac{62}{10} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{45}{10} + \frac{62}{10} = \frac{107}{10} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 107 by 10 to get the quotient and remainder: [tex]\(107 \div 10 = 10\)[/tex] remainder [tex]\(7\)[/tex]
- So, [tex]\( \frac{107}{10} = 10 \frac{7}{10}\)[/tex]
Final result: [tex]\( 4 \frac{1}{2} + 6 \frac{1}{5} = 10 \frac{7}{10} \)[/tex]
### Problem 2: [tex]\(2 \frac{5}{10} + 5 \frac{3}{5}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(2 \frac{5}{10} = \frac{2 \cdot 10 + 5}{10} = \frac{25}{10}\)[/tex]
- [tex]\(5 \frac{3}{5} = \frac{5 \cdot 5 + 3}{5} = \frac{28}{5}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 10 and 5 is 50.
- Convert [tex]\( \frac{25}{10} \)[/tex] to have a denominator of 50: [tex]\( \frac{25 \cdot 5}{10 \cdot 5} = \frac{125}{50} \)[/tex]
- Convert [tex]\( \frac{28}{5} \)[/tex] to have a denominator of 50: [tex]\( \frac{28 \cdot 10}{5 \cdot 10} = \frac{280}{50} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{125}{50} + \frac{280}{50} = \frac{405}{50} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 405 by 50 to get the quotient and remainder: [tex]\(405 \div 50 = 8\)[/tex] remainder [tex]\(5\)[/tex]
- So, [tex]\( \frac{405}{50} = 8 \frac{5}{50}\)[/tex]
Final result: [tex]\( 2 \frac{5}{10} + 5 \frac{3}{5} = 8 \frac{5}{50} \)[/tex]
### Problem 3: [tex]\(3 \frac{4}{5} + 8 \frac{1}{3}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(3 \frac{4}{5} = \frac{3 \cdot 5 + 4}{5} = \frac{19}{5}\)[/tex]
- [tex]\(8 \frac{1}{3} = \frac{8 \cdot 3 + 1}{3} = \frac{25}{3}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 5 and 3 is 15.
- Convert [tex]\( \frac{19}{5} \)[/tex] to have a denominator of 15: [tex]\( \frac{19 \cdot 3}{5 \cdot 3} = \frac{57}{15} \)[/tex]
- Convert [tex]\( \frac{25}{3} \)[/tex] to have a denominator of 15: [tex]\( \frac{25 \cdot 5}{3 \cdot 5} = \frac{125}{15} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{57}{15} + \frac{125}{15} = \frac{182}{15} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 182 by 15 to get the quotient and remainder: [tex]\(182 \div 15 = 12\)[/tex] remainder [tex]\(2\)[/tex]
- So, [tex]\( \frac{182}{15} = 12 \frac{2}{15}\)[/tex]
Final result: [tex]\( 3 \frac{4}{5} + 8 \frac{1}{3} = 12 \frac{2}{15} \)[/tex]
### Problem 4: [tex]\(2 \frac{4}{5} + 8 \frac{3}{4}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(2 \frac{4}{5} = \frac{2 \cdot 5 + 4}{5} = \frac{14}{5}\)[/tex]
- [tex]\(8 \frac{3}{4} = \frac{8 \cdot 4 + 3}{4} = \frac{35}{4}\)[/tex]
2. Find a common denominator and convert fractions:
- The common denominator between 5 and 4 is 20.
- Convert [tex]\( \frac{14}{5} \)[/tex] to have a denominator of 20: [tex]\( \frac{14 \cdot 4}{5 \cdot 4} = \frac{56}{20} \)[/tex]
- Convert [tex]\( \frac{35}{4} \)[/tex] to have a denominator of 20: [tex]\( \frac{35 \cdot 5}{4 \cdot 5} = \frac{175}{20} \)[/tex]
3. Add the fractions:
- [tex]\( \frac{56}{20} + \frac{175}{20} = \frac{231}{20} \)[/tex]
4. Convert the improper fraction back to a mixed number:
- Divide 231 by 20 to get the quotient and remainder: [tex]\(231 \div 20 = 11\)[/tex] remainder [tex]\(11\)[/tex]
- So, [tex]\( \frac{231}{20} = 11 \frac{11}{20}\)[/tex]
Final result: [tex]\( 2 \frac{4}{5} + 8 \frac{3}{4} = 11 \frac{11}{20} \)[/tex]
Thus, the solved problems are:
1. [tex]\( 4 \frac{1}{2} + 6 \frac{1}{5} = 10 \frac{7}{10} \)[/tex]
2. [tex]\( 2 \frac{5}{10} + 5 \frac{3}{5} = 8 \frac{5}{50} \)[/tex]
3. [tex]\( 3 \frac{4}{5} + 8 \frac{1}{3} = 12 \frac{2}{15} \)[/tex]
4. [tex]\( 2 \frac{4}{5} + 8 \frac{3}{4} = 11 \frac{11}{20} \)[/tex]
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