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Number foundations · Topic 3 of 15

Operations with Fractions

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Math · fractions

Operations with Fractions

Add and subtract over a common denominator, multiply straight across, and divide by multiplying by the reciprocal.

01 · MultiplyMultiply straight across, then simplify
02 · Add or subtractAdd/subtract: common denominator first
03 · DivideDivide = multiply by the reciprocal
01

The rules

Add or subtract the numerators over a common denominator must know
Unlike denominators: rewrite both over the LCD first must know
Multiply: numerator × numerator, denominator × denominator must know
Divide: multiply by the reciprocal of the second fraction must know
Mixed numbers × or ÷: change to improper fractions first (1 1/2 = 3/2)
Mixed + or −: wholes and fractions separately, or go improper
A whole number is itself over 1 (5 = 5/1)
02

Adding with unlike denominators

Find the LCD: count up multiples of the larger denominator
Rebuild each fraction over the LCD (top and bottom × the same number)
Add the numerators, keep the LCD, then simplify
03

Worked examples

2/5 + 1/3 = 6/15 + 5/15 = 11/15
3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 1 1/2
"1/2 of 3/4 cup" = 1/2 × 3/4 = 3/8 cup (of = multiply)
04

Adding or multiplying fractions?

Commonly confused
Add / subtractMultiply / divide
Common denominator needed?Yes — rewrite over the LCD firstNo
DenominatorsStay the sameMultiply them (divide: flip first)
Mixed numbersCan work with whole parts separatelyChange to improper fractions first
Example1/3 + 1/4 = 4/12 + 3/12 = 7/121/3 × 1/4 = 1/12
05

Common mistakes

Never add the denominators: 1/3 + 1/4 = 7/12, not 2/7
06

More detail

Show 4 more facts
Reciprocal = the fraction flipped (4/7 → 7/4); 0 has none
Flip only the second fraction (the one you divide by)
Cross-cancel before multiplying — same answer, smaller numbers
Subtracting mixed numbers: borrow 1 from the whole if the fraction is too small
Pearl

Keep, change, flip: keep the first fraction, change ÷ to ×, flip the second.

Sources · OpenStax Prealgebra 2e §4.2 Multiply and Divide Fractions · OpenStax Prealgebra 2e §4.3 Multiply and Divide Mixed Numbers · OpenStax Prealgebra 2e §4.4 Add and Subtract Fractions with Common Denominators · OpenStax Prealgebra 2e §4.5 Add and Subtract Fractions with Different Denominators · OpenStax Prealgebra 2e §4.6 Add and Subtract Mixed Numbers · Union Test Prep — HESI A2 Mathematics study guide, p. 2: Fractions · Built from the verified NurseSavvy HESI A2 fact map (facts reviewed 2026-10-16). HESI is a trademark of Elsevier; NurseSavvy is not affiliated with Elsevier.

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