Add \( \frac{2}{3} + \frac{1}{4} \).
Find a common denominator: \( \frac{2}{3} = \frac{8}{12} \) and \( \frac{1}{4} = \frac{3}{12} \).
Add: \( \frac{8}{12} + \frac{3}{12} = \frac{11}{12} \).
Subtract \( 1\frac{1}{2} – \frac{3}{4} \).
Convert mixed number: \( 1\frac{1}{2} = \frac{3}{2} \).
Find a common denominator: \( \frac{3}{2} = \frac{6}{4} \).
Subtract: \( \frac{6}{4} – \frac{3}{4} = \frac{3}{4} \).
The reciprocal of \( \frac{1}{5} \) is \( 5 \) because \( \frac{1}{5} \times 5 = 1 \).
Use the multiplicative inverse to solve \( \frac{1}{5} \times x = 3 \).
\( x = 3 \times 5 = 15 \).
Multiply \( \frac{2}{3} \times \frac{4}{5} \).
Multiply the numerators and denominators: \( \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \).
Divide \( \frac{7}{8} \div \frac{2}{3} \).
Multiply by the reciprocal: \( \frac{7}{8} \times \frac{3}{2} = \frac{7 \times 3}{8 \times 2} = \frac{21}{16} \).
Multiply mixed numbers: \( 1\frac{1}{2} \times 2\frac{2}{3} \).
Convert to improper fractions: \( \frac{3}{2} \times \frac{8}{3} = \frac{3 \times 8}{2 \times 3} = 4 \).
To add \( \frac{2}{3} + \frac{1}{4} \), find the LCM of 3 and 4, which is 12.
Convert \( \frac{2}{3} \) to \( \frac{8}{12} \) and \( \frac{1}{4} \) to \( \frac{3}{12} \).
Add the fractions: \( \frac{8}{12} + \frac{3}{12} = \frac{11}{12} \).
The multiplicative inverse of \( \frac{1}{5} \) is 5 because \( \frac{1}{5} \times 5 = 1 \).
To solve \( \frac{1}{5} \times x = 3 \), multiply both sides by 5:
\( x = 3 \times 5 = 15 \).
Multiply \( \frac{2}{3} \times \frac{4}{5} \) by multiplying the numerators and denominators:
\( \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \).
Divide \( \frac{7}{8} \div \frac{2}{3} \) by multiplying by the reciprocal:
\( \frac{7}{8} \times \frac{3}{2} = \frac{21}{16} \).
Incorrect: Adding \( \frac{2}{3} + \frac{1}{4} \) without finding a common denominator.
Correct: Finding the common denominator (12) and converting: \( \frac{2}{3} = \frac{8}{12} \) and \( \frac{1}{4} = \frac{3}{12} \).
Then add: \( \frac{8}{12} + \frac{3}{12} = \frac{11}{12} \).
Incorrect: Identifying the multiplicative inverse of \( \frac{1}{5} \) as \( \frac{1}{5} \).
Correct: The multiplicative inverse of \( \frac{1}{5} \) is 5.
To solve \( \frac{1}{5} \times x = 3 \), multiply both sides by 5:
\( x = 3 \times 5 = 15 \).
Incorrect: Multiplying \( \frac{2}{3} \times \frac{4}{5} \) by adding the numerators and denominators: \( \frac{2 + 4}{3 + 5} = \frac{6}{8} \).
Correct: Multiply the numerators and denominators: \( \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \).
Incorrect: Dividing mixed numbers without converting: \( 1\frac{1}{2} \div 2\frac{2}{3} \).
Correct: Convert to improper fractions first: \( \frac{3}{2} \div \frac{8}{3} = \frac{3}{2} \times \frac{3}{8} = \frac{9}{16} \).
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