Fractions: Operations and Mixed Numbers

Fractions name equal parts

A denominator tells how many equal parts make one whole; a numerator tells how many parts are selected. Equivalent fractions name the same quantity, such as 1/2 and 3/6. Multiply or divide numerator and denominator by the same nonzero number to rename without changing value. A fraction is simplified when numerator and denominator share no factor greater than 1.

Different pieces, same amount

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Addition and subtraction

Fractions can be added only when their pieces have the same size. For 1/3+1/4, the least common denominator is 12: 4/12+3/12=7/12. Do not add denominators; twelfths remain twelfths. For subtraction, use the same process. If borrowing is needed, rename one whole as a fraction with the chosen denominator.

Multiply, divide, and convert

To multiply, multiply numerators and multiply denominators; cancel common factors first when convenient. To divide, multiply by the reciprocal of the divisor. Only the second fraction flips. A mixed number such as 2 1/3 means 2+1/3. Convert it to an improper fraction: (2×3+1)/3=7/3. Convert back by division.

Estimate with benchmarks 0, 1/2, and 1. A product of two positive proper fractions must be smaller than either factor. A quotient by a positive fraction less than 1 must be larger than the dividend.

Worked examples

Worked example 1: 2 1/4−5/6. Convert 2 1/4 to 9/4. The common denominator is 12: 27/12−10/12=17/12=1 5/12. The estimate 2.25−0.83≈1.42 supports the result.

Worked example 2: 1 1/2÷3/8. Convert to 3/2 and multiply by 8/3. Cancel the 3s: 8/2=4. Dividing 1.5 by a number below 1 should increase it, so 4 is reasonable.

A reliable way to reason

The least common denominator is efficient, but any common denominator works. For 5/6−1/4, using 12 gives 10/12−3/12=7/12; using 24 gives 20/24−6/24=14/24=7/12.

Division asks how many groups fit: 3÷3/4=4 because four groups of three fourths make 3. For mixed numbers, convert before multiplying or dividing; for addition or subtraction, keeping mixed numbers can be convenient unless borrowing is required.

Guided reasoning and error checks

For a multi-operation fraction expression, write one transformation per line. Consider 2/3+(3/4×2/5). Multiply first: 3/4×2/5=6/20=3/10. Then rename for addition: 2/3=20/30 and 3/10=9/30, so the total is 29/30. An answer above 1 would be suspicious because the estimate 2/3+3/10 is just under 1. This workflow separates the multiplication rule from the addition rule and makes the denominator choice visible.

Your quick summary

• Rename unlike fractions with a common denominator.
• Cross-cancel before multiplying when possible.
• Use a reciprocal only for division, and only on the divisor.
• Use benchmark fractions to check whether a sum, product, or quotient has a reasonable size.