Math · Numbers, fractions & decimals
Fractions: Operations and Mixed Numbers
Estimated study time. Take the time you need.
Start here: key ideas
- Add or subtract only after naming equal-size parts.
- Multiply numerators and denominators, then simplify.
- Divide by multiplying by the reciprocal.
- Convert mixed numbers before multiplying or dividing.
- Estimate before accepting an answer.
What you’ll be able to do
- Find common denominators and simplify.
- Multiply, divide, and convert mixed numbers.
- Estimate to test reasonableness.
Use denominator meaning to choose the operation, then simplify and estimate.
No prior lesson is required; multiplication facts are reviewed within the examples.
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.
Denny Burzynski and Wade Ellis, Jr. / OpenStax CNX, via Mathematics LibreTexts · CC BY · https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/04%3A_Introduction_to_Fractions_and_Multiplication_and_Division_of_Fractions/4.03%3A_Equivalent_Fractions_Reducing_Fractions_to_Lowest_Terms_and_Raising_Fractions_to_Higher_Terms · https://creativecommons.org/licenses/by/ · Source image reproduced unchanged.
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.
Key terms
- numerator: The number of selected equal parts.
- denominator: The number of equal parts in one whole.
- equivalent fractions: Fractions naming the same value.
- least common denominator: The least common multiple of the denominators.
- mixed number: A whole number plus a proper fraction.
- improper fraction: A fraction whose numerator is at least its denominator.
- reciprocal: A fraction formed by switching numerator and denominator.
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.
Add needs same-size pieces; divide means multiply by the flipped second fraction.
Sources
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