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Sagot :
Sure! Let me guide you through each problem step-by-step:
### Problem 1: [tex]\(4 \frac{1}{2} - 5 \frac{1}{4}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2}\)[/tex]
- [tex]\(5 \frac{1}{4} = 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}\)[/tex]
2. Find a common denominator and subtract:
- [tex]\(\frac{9}{2} = \frac{18}{4}\)[/tex]
Then, [tex]\(\frac{18}{4} - \frac{21}{4} = \frac{18 - 21}{4} = -\frac{3}{4}\)[/tex]
So, [tex]\(4 \frac{1}{2} - 5 \frac{1}{4} = -\frac{3}{4} = -0.75\)[/tex]
### Problem 2: [tex]\(2 \cdot 5 \frac{2}{3} - 3 \frac{1}{6}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(5 \frac{2}{3} = 5 + \frac{2}{3} = \frac{15}{3} + \frac{2}{3} = \frac{17}{3}\)[/tex]
- [tex]\(3 \frac{1}{6} = 3 + \frac{1}{6} = \frac{18}{6} + \frac{1}{6} = \frac{19}{6}\)[/tex]
2. Multiply and then subtract:
- [tex]\(2 \cdot \frac{17}{3} = \frac{34}{3}\)[/tex]
Then, find a common denominator for subtraction:
- [tex]\(\frac{34}{3} = \frac{68}{6}\)[/tex]
Then, [tex]\(\frac{68}{6} - \frac{19}{6} = \frac{68 - 19}{6} = \frac{49}{6}\)[/tex]
3. Convert back to a mixed number:
- [tex]\(\frac{49}{6} = 8 \frac{1}{6}\)[/tex]
So, [tex]\(2 \cdot 5 \frac{2}{3} - 3 \frac{1}{6} = 8 \frac{1}{6} \approx 8.166666666666668\)[/tex]
### Problem 3: [tex]\(3 \frac{6}{9} - 2 \frac{1}{3}\)[/tex]
1. Simplify fractions:
- [tex]\(3 \frac{6}{9} = 3 + \frac{6}{9} = 3 + \frac{2}{3}\)[/tex]
2. Convert the mixed numbers:
- [tex]\(3 \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3}\)[/tex]
- [tex]\(2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)[/tex]
3. Subtract:
- [tex]\(\frac{11}{3} - \frac{7}{3} = \frac{11 - 7}{3} = \frac{4}{3}\)[/tex]
4. Convert back to a mixed number:
- [tex]\(\frac{4}{3} = 1 \frac{1}{3}\)[/tex]
So, [tex]\(3 \frac{6}{9} - 2 \frac{1}{3} = 1 \frac{1}{3} = 1.333333333333333\)[/tex]
### Problem 4: [tex]\(3 \frac{1}{2} - 2 \frac{1}{3}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\)[/tex]
- [tex]\(2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)[/tex]
2. Find a common denominator and subtract:
- [tex]\(\frac{7}{2} = \frac{21}{6}\)[/tex]
- [tex]\(\frac{7}{3} = \frac{14}{6}\)[/tex]
Then, [tex]\( \frac{21}{6} - \frac{14}{6} = \frac{7}{6}\)[/tex]
3. Convert back to a mixed number:
- [tex]\(\frac{7}{6} = 1 \frac{1}{6} \approx 1.1666666666666665\)[/tex]
So, [tex]\(3 \frac{1}{2} - 2 \frac{1}{3} = 1 \frac{1}{6} \approx 1.1666666666666665\)[/tex]
### Problem 5: [tex]\(4 \frac{3}{4} - 2 \frac{2}{5}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(4 \frac{3}{4} = 4 + \frac{3}{4} = \frac{16}{4} + \frac{3}{4} = \frac{19}{4}\)[/tex]
- [tex]\(2 \frac{2}{5} = 2 + \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5}\)[/tex]
2. Find a common denominator and subtract:
- [tex]\(\frac{19}{4}\)[/tex] and [tex]\(\frac{12}{5}\)[/tex] common denominator is 20:
- [tex]\(\frac{19}{4} = \frac{95}{20}\)[/tex]
- [tex]\(\frac{12}{5} = \frac{48}{20}\)[/tex]
Then, [tex]\(\frac{95}{20} - \frac{48}{20} = \frac{95 - 48}{20} = \frac{47}{20}\)[/tex]
3. Convert back to a mixed number:
- [tex]\(\frac{47}{20} = 2 \frac{7}{20}\)[/tex]
So, [tex]\(4 \frac{3}{4} - 2 \frac{2}{5} = 2 \frac{7}{20} \approx 2.35\)[/tex]
### Problem 1: [tex]\(4 \frac{1}{2} - 5 \frac{1}{4}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2}\)[/tex]
- [tex]\(5 \frac{1}{4} = 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}\)[/tex]
2. Find a common denominator and subtract:
- [tex]\(\frac{9}{2} = \frac{18}{4}\)[/tex]
Then, [tex]\(\frac{18}{4} - \frac{21}{4} = \frac{18 - 21}{4} = -\frac{3}{4}\)[/tex]
So, [tex]\(4 \frac{1}{2} - 5 \frac{1}{4} = -\frac{3}{4} = -0.75\)[/tex]
### Problem 2: [tex]\(2 \cdot 5 \frac{2}{3} - 3 \frac{1}{6}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(5 \frac{2}{3} = 5 + \frac{2}{3} = \frac{15}{3} + \frac{2}{3} = \frac{17}{3}\)[/tex]
- [tex]\(3 \frac{1}{6} = 3 + \frac{1}{6} = \frac{18}{6} + \frac{1}{6} = \frac{19}{6}\)[/tex]
2. Multiply and then subtract:
- [tex]\(2 \cdot \frac{17}{3} = \frac{34}{3}\)[/tex]
Then, find a common denominator for subtraction:
- [tex]\(\frac{34}{3} = \frac{68}{6}\)[/tex]
Then, [tex]\(\frac{68}{6} - \frac{19}{6} = \frac{68 - 19}{6} = \frac{49}{6}\)[/tex]
3. Convert back to a mixed number:
- [tex]\(\frac{49}{6} = 8 \frac{1}{6}\)[/tex]
So, [tex]\(2 \cdot 5 \frac{2}{3} - 3 \frac{1}{6} = 8 \frac{1}{6} \approx 8.166666666666668\)[/tex]
### Problem 3: [tex]\(3 \frac{6}{9} - 2 \frac{1}{3}\)[/tex]
1. Simplify fractions:
- [tex]\(3 \frac{6}{9} = 3 + \frac{6}{9} = 3 + \frac{2}{3}\)[/tex]
2. Convert the mixed numbers:
- [tex]\(3 \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3}\)[/tex]
- [tex]\(2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)[/tex]
3. Subtract:
- [tex]\(\frac{11}{3} - \frac{7}{3} = \frac{11 - 7}{3} = \frac{4}{3}\)[/tex]
4. Convert back to a mixed number:
- [tex]\(\frac{4}{3} = 1 \frac{1}{3}\)[/tex]
So, [tex]\(3 \frac{6}{9} - 2 \frac{1}{3} = 1 \frac{1}{3} = 1.333333333333333\)[/tex]
### Problem 4: [tex]\(3 \frac{1}{2} - 2 \frac{1}{3}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\)[/tex]
- [tex]\(2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\)[/tex]
2. Find a common denominator and subtract:
- [tex]\(\frac{7}{2} = \frac{21}{6}\)[/tex]
- [tex]\(\frac{7}{3} = \frac{14}{6}\)[/tex]
Then, [tex]\( \frac{21}{6} - \frac{14}{6} = \frac{7}{6}\)[/tex]
3. Convert back to a mixed number:
- [tex]\(\frac{7}{6} = 1 \frac{1}{6} \approx 1.1666666666666665\)[/tex]
So, [tex]\(3 \frac{1}{2} - 2 \frac{1}{3} = 1 \frac{1}{6} \approx 1.1666666666666665\)[/tex]
### Problem 5: [tex]\(4 \frac{3}{4} - 2 \frac{2}{5}\)[/tex]
1. Convert mixed numbers to improper fractions:
- [tex]\(4 \frac{3}{4} = 4 + \frac{3}{4} = \frac{16}{4} + \frac{3}{4} = \frac{19}{4}\)[/tex]
- [tex]\(2 \frac{2}{5} = 2 + \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5}\)[/tex]
2. Find a common denominator and subtract:
- [tex]\(\frac{19}{4}\)[/tex] and [tex]\(\frac{12}{5}\)[/tex] common denominator is 20:
- [tex]\(\frac{19}{4} = \frac{95}{20}\)[/tex]
- [tex]\(\frac{12}{5} = \frac{48}{20}\)[/tex]
Then, [tex]\(\frac{95}{20} - \frac{48}{20} = \frac{95 - 48}{20} = \frac{47}{20}\)[/tex]
3. Convert back to a mixed number:
- [tex]\(\frac{47}{20} = 2 \frac{7}{20}\)[/tex]
So, [tex]\(4 \frac{3}{4} - 2 \frac{2}{5} = 2 \frac{7}{20} \approx 2.35\)[/tex]
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