5.3 Fraction Addition and Subtraction

Fraction Addition and Subtraction are fundamental operations in elementary algebra that involve combining or subtracting fractions with like or unlike denominators.

Fraction Addition and Subtraction covers the procedures for combining or finding the difference between two or more fractions, distinguishing the direct case where denominators already match from the case requiring a common denominator first, extending these procedures to signed fractions and mixed numbers, and describing the simplification and verification steps that complete the process.

Addition with Equal Denominators

Addition with equal denominators adds the numerators of two or more fractions directly while keeping the shared denominator unchanged, such as 2/7 + 3/7 = 5/7.

\frac{2}{7} + \frac{3}{7} = \frac{5}{7}

Subtraction with Equal Denominators

Subtraction with equal denominators subtracts the numerators of two fractions directly while keeping the shared denominator unchanged, such as 5/7 − 3/7 = 2/7.

\frac{5}{7} - \frac{3}{7} = \frac{2}{7}

Addition with Unequal Denominators

Addition with unequal denominators first rewrites both fractions as equivalent fractions sharing a common denominator, then adds their numerators as in the equal-denominator case, such as 1/2 + 1/3 rewritten as 3/6 + 2/6, giving 5/6.

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Subtraction with Unequal Denominators

Subtraction with unequal denominators first rewrites both fractions as equivalent fractions sharing a common denominator, then subtracts their numerators as in the equal-denominator case, such as 1/2 − 1/3 rewritten as 3/6 − 2/6, giving 1/6.

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

Signed Fraction Addition

Signed fraction addition applies the same signed-addition rules established for whole numbers—matching-sign addends combine magnitudes directly, opposite-sign addends combine through subtraction of magnitudes—to the numerators of fractions after a common denominator has been established.

- \frac{1}{4} + \frac{3}{4} = \frac{2}{4} = \frac{1}{2}

Signed Fraction Subtraction

Signed fraction subtraction rewrites the subtraction as addition of the opposite fraction, then applies signed fraction addition, following exactly the same reformulation already established for signed whole-number subtraction.

\frac{1}{4} - \frac{3}{4} = \frac{1}{4} + (- \frac{3}{4})

Mixed Number Addition

Mixed number addition combines the whole-number parts and the fractional parts of two mixed numbers separately, then combines those two partial results together, converting the fractional parts to a common denominator first whenever needed.

2 \frac{1}{3} + 1 \frac{1}{3} = 3 \frac{2}{3}

Mixed Number Subtraction

Mixed number subtraction subtracts the whole-number parts and the fractional parts of two mixed numbers separately, converting the fractional parts to a common denominator first whenever needed, and requiring regrouping whenever the fractional part being subtracted is larger than the fractional part it is being subtracted from.

Mixed Number Regrouping

Mixed number regrouping converts one whole unit from a mixed number's whole-number part into an equivalent fraction added to its fractional part, performed whenever a mixed number subtraction would otherwise require subtracting a larger fractional part from a smaller one.

3 \frac{1}{4} = 2 \frac{5}{4}

Fraction Result Simplification

Fraction result simplification reduces a computed sum or difference to simplest form and, if the result is improper, optionally converts it into a mixed number, completing the fraction addition or subtraction process with a fully reduced final answer.

Fraction Sum and Difference Verification

Fraction sum and difference verification confirms a computed result by converting all fractions involved into decimal form and checking that the decimal values are consistent with the fraction-based computation, or by applying the appropriate inverse operation to the computed result and confirming it returns one of the original fractions.

Together, these procedures establish a complete method for adding and subtracting fractions: matching denominators either directly or through common denominator construction, applying signed-number rules to the resulting numerators, handling mixed numbers through separate whole-and-fractional-part combination with regrouping as needed, and finishing with simplification and verification of the final result.