Add, subtract, multiply, and divide fractions fluently with common denominators.
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Add and subtract over a common denominator. Use the least common denominator, not just the product: for \(\frac{1}{3}\) and \(\frac{1}{9}\), use 9, not 27.
Common slips: adding the denominators (\(\frac{1}{2} + \frac{1}{3}\) is \(\frac{5}{6}\), not \(\frac{2}{5}\)), losing the whole-number part of a mixed number, and sign errors with negatives. Estimate first: \(1\frac{3}{7} + 1\frac{3}{4}\) is a bit more than 3.
Unit rate \(= \frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}\) cups. (Dividing the other way gives 8/9. Multiplying the two fractions gives 1/2. Flipping both fractions and then multiplying gives 2.)
Convert to a common denominator of 9: \(-\frac{1}{3} - 1\frac{1}{9} = -1\frac{4}{9}\). Subtracting the numerators and adding the denominators gives \(-\frac{11}{12}\). Subtracting in the wrong order gives \(1\frac{4}{9}\). Subtracting the numerators but keeping the first denominator gives \(-3\frac{2}{3}\).
Convert to a common denominator of 77: \(\frac{2}{7} + \frac{8}{11} = 1\frac{1}{77}\). Adding the numerators and adding the denominators gives \(\frac{5}{9}\). Subtracting instead of adding gives \(-\frac{34}{77}\). Adding the numerators but keeping the first denominator gives \(1\frac{3}{7}\).