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Of course! Let's solve each fraction problem step-by-step, assuming we need to perform the operations manually for a detailed understanding.
### Problem 1: [tex]\( 2 \frac{1}{3} + 1 \frac{2}{9} \)[/tex]
#### Step 1: Convert mixed numbers to improper fractions
- [tex]\( 2 \frac{1}{3} = \frac{2 \cdot 3 + 1}{3} = \frac{7}{3} \)[/tex]
- [tex]\( 1 \frac{2}{9} = \frac{1 \cdot 9 + 2}{9} = \frac{11}{9} \)[/tex]
#### Step 2: Find a common denominator
- The denominators are 3 and 9. The least common multiple (LCM) of 3 and 9 is 9.
#### Step 3: Adjust fractions to have the common denominator
- [tex]\( \frac{7}{3} = \frac{7 \cdot 3}{3 \cdot 3} = \frac{21}{9} \)[/tex]
- [tex]\( \frac{11}{9} \)[/tex] remains the same
#### Step 4: Add the fractions
- [tex]\( \frac{21}{9} + \frac{11}{9} = \frac{21 + 11}{9} = \frac{32}{9} \)[/tex]
#### Step 5: Convert the improper fraction back to a mixed number (if needed)
- [tex]\( \frac{32}{9} = 3 \frac{5}{9} \)[/tex]
Thus, [tex]\( 2 \frac{1}{3} + 1 \frac{2}{9} = 3 \frac{5}{9} \)[/tex].
### Problem 2: [tex]\( 2 \cdot \frac{1}{2} - \frac{1}{6} \)[/tex]
#### Step 1: Convert the whole division and fraction to an improper fraction
- Convert [tex]\( 2 \cdot \frac{1}{2} \)[/tex]: [tex]\( 2 \cdot \frac{1}{2} = \frac{4}{2} = 2 \)[/tex]
#### Step 2: Find the common denominator for the subtraction
- The denominators are 2 and 6. The least common multiple (LCM) of 2 and 6 is 6.
#### Step 3: Adjust fractions to have a common denominator
- [tex]\( 2 = \frac{2 \cdot 3}{1 \cdot 3} = \frac{6}{3} = \frac{12}{6} \)[/tex]
- [tex]\( \frac{1}{6} \)[/tex] remains the same
#### Step 4: Subtract the fractions
- [tex]\( \frac{12}{6} - \frac{1}{6} = \frac{12 - 1}{6} = \frac{11}{6} \)[/tex]
#### Step 5: Convert the improper fraction back to a mixed number (if needed)
- [tex]\( \frac{11}{6} = 1 \frac{5}{6} \)[/tex]
Thus, [tex]\( 2 \cdot \frac{1}{2} - \frac{1}{6} = 1 \frac{5}{6} \)[/tex].
### Problem 3: [tex]\( 1 \frac{7}{8} - \frac{1}{4} \)[/tex]
#### Step 1: Convert mixed number to an improper fraction
- [tex]\( 1 \frac{7}{8} = \frac{1 \cdot 8 + 7}{8} = \frac{15}{8} \)[/tex]
#### Step 2: Find a common denominator for subtraction
- The denominators are 8 and 4. The least common multiple (LCM) of 8 and 4 is 8.
#### Step 3: Adjust fractions to have the common denominator
- [tex]\( \frac{15}{8} \)[/tex] remains the same
- [tex]\( \frac{1}{4} = \frac{1 \cdot 2}{4 \cdot 2} = \frac{2}{8} \)[/tex]
#### Step 4: Subtract the fractions
- [tex]\( \frac{15}{8} - \frac{2}{8} = \frac{15 - 2}{8} = \frac{13}{8} \)[/tex]
#### Step 5: Convert the improper fraction back to a mixed number (if needed)
- [tex]\( \frac{13}{8} = 1 \frac{5}{8} \)[/tex]
Thus, [tex]\( 1 \frac{7}{8} - \frac{1}{4} = 1 \frac{5}{8} \)[/tex].
### Problem 4: [tex]\( \frac{2}{5} + \frac{3}{10} \)[/tex]
#### Step 1: Find a common denominator for addition
- The denominators are 5 and 10. The least common multiple (LCM) of 5 and 10 is 10.
#### Step 2: Adjust fractions to have the common denominator
- [tex]\( \frac{2}{5} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10} \)[/tex]
- [tex]\( \frac{3}{10} \)[/tex] remains the same
#### Step 3: Add the fractions
- [tex]\( \frac{4}{10} + \frac{3}{10} = \frac{4 + 3}{10} = \frac{7}{10} \)[/tex]
Thus, [tex]\( \frac{2}{5} + \frac{3}{10} = \frac{7}{10} \)[/tex].
In summary:
1. [tex]\( 2 \frac{1}{3} + 1 \frac{2}{9} = 3 \frac{5}{9} \)[/tex]
2. [tex]\( 2 \cdot \frac{1}{2} - \frac{1}{6} = 1 \frac{5}{6} \)[/tex]
3. [tex]\( 1 \frac{7}{8} - \frac{1}{4} = 1 \frac{5}{8} \)[/tex]
4. [tex]\( \frac{2}{5} + \frac{3}{10} = \frac{7}{10} \)[/tex]
### Problem 1: [tex]\( 2 \frac{1}{3} + 1 \frac{2}{9} \)[/tex]
#### Step 1: Convert mixed numbers to improper fractions
- [tex]\( 2 \frac{1}{3} = \frac{2 \cdot 3 + 1}{3} = \frac{7}{3} \)[/tex]
- [tex]\( 1 \frac{2}{9} = \frac{1 \cdot 9 + 2}{9} = \frac{11}{9} \)[/tex]
#### Step 2: Find a common denominator
- The denominators are 3 and 9. The least common multiple (LCM) of 3 and 9 is 9.
#### Step 3: Adjust fractions to have the common denominator
- [tex]\( \frac{7}{3} = \frac{7 \cdot 3}{3 \cdot 3} = \frac{21}{9} \)[/tex]
- [tex]\( \frac{11}{9} \)[/tex] remains the same
#### Step 4: Add the fractions
- [tex]\( \frac{21}{9} + \frac{11}{9} = \frac{21 + 11}{9} = \frac{32}{9} \)[/tex]
#### Step 5: Convert the improper fraction back to a mixed number (if needed)
- [tex]\( \frac{32}{9} = 3 \frac{5}{9} \)[/tex]
Thus, [tex]\( 2 \frac{1}{3} + 1 \frac{2}{9} = 3 \frac{5}{9} \)[/tex].
### Problem 2: [tex]\( 2 \cdot \frac{1}{2} - \frac{1}{6} \)[/tex]
#### Step 1: Convert the whole division and fraction to an improper fraction
- Convert [tex]\( 2 \cdot \frac{1}{2} \)[/tex]: [tex]\( 2 \cdot \frac{1}{2} = \frac{4}{2} = 2 \)[/tex]
#### Step 2: Find the common denominator for the subtraction
- The denominators are 2 and 6. The least common multiple (LCM) of 2 and 6 is 6.
#### Step 3: Adjust fractions to have a common denominator
- [tex]\( 2 = \frac{2 \cdot 3}{1 \cdot 3} = \frac{6}{3} = \frac{12}{6} \)[/tex]
- [tex]\( \frac{1}{6} \)[/tex] remains the same
#### Step 4: Subtract the fractions
- [tex]\( \frac{12}{6} - \frac{1}{6} = \frac{12 - 1}{6} = \frac{11}{6} \)[/tex]
#### Step 5: Convert the improper fraction back to a mixed number (if needed)
- [tex]\( \frac{11}{6} = 1 \frac{5}{6} \)[/tex]
Thus, [tex]\( 2 \cdot \frac{1}{2} - \frac{1}{6} = 1 \frac{5}{6} \)[/tex].
### Problem 3: [tex]\( 1 \frac{7}{8} - \frac{1}{4} \)[/tex]
#### Step 1: Convert mixed number to an improper fraction
- [tex]\( 1 \frac{7}{8} = \frac{1 \cdot 8 + 7}{8} = \frac{15}{8} \)[/tex]
#### Step 2: Find a common denominator for subtraction
- The denominators are 8 and 4. The least common multiple (LCM) of 8 and 4 is 8.
#### Step 3: Adjust fractions to have the common denominator
- [tex]\( \frac{15}{8} \)[/tex] remains the same
- [tex]\( \frac{1}{4} = \frac{1 \cdot 2}{4 \cdot 2} = \frac{2}{8} \)[/tex]
#### Step 4: Subtract the fractions
- [tex]\( \frac{15}{8} - \frac{2}{8} = \frac{15 - 2}{8} = \frac{13}{8} \)[/tex]
#### Step 5: Convert the improper fraction back to a mixed number (if needed)
- [tex]\( \frac{13}{8} = 1 \frac{5}{8} \)[/tex]
Thus, [tex]\( 1 \frac{7}{8} - \frac{1}{4} = 1 \frac{5}{8} \)[/tex].
### Problem 4: [tex]\( \frac{2}{5} + \frac{3}{10} \)[/tex]
#### Step 1: Find a common denominator for addition
- The denominators are 5 and 10. The least common multiple (LCM) of 5 and 10 is 10.
#### Step 2: Adjust fractions to have the common denominator
- [tex]\( \frac{2}{5} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10} \)[/tex]
- [tex]\( \frac{3}{10} \)[/tex] remains the same
#### Step 3: Add the fractions
- [tex]\( \frac{4}{10} + \frac{3}{10} = \frac{4 + 3}{10} = \frac{7}{10} \)[/tex]
Thus, [tex]\( \frac{2}{5} + \frac{3}{10} = \frac{7}{10} \)[/tex].
In summary:
1. [tex]\( 2 \frac{1}{3} + 1 \frac{2}{9} = 3 \frac{5}{9} \)[/tex]
2. [tex]\( 2 \cdot \frac{1}{2} - \frac{1}{6} = 1 \frac{5}{6} \)[/tex]
3. [tex]\( 1 \frac{7}{8} - \frac{1}{4} = 1 \frac{5}{8} \)[/tex]
4. [tex]\( \frac{2}{5} + \frac{3}{10} = \frac{7}{10} \)[/tex]
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