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Lowest Terms: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Lowest TermsLowest Terms: Definition and ExampleTable of ContentsDefinition of Lowest Terms

A fraction is said to be in lowest terms (or simplest form) when the numerator and denominator have no common factors other than 111. For example, 78\frac{7}{8}87​ is in lowest terms because 777 and 888 share no common factors. However, 68\frac{6}{8}86​ is not in lowest terms since both 666 and 888 are divisible by 222. Mathematically speaking, a fraction ab\frac{a}{b}ba​ (where b ≠ 000) is in lowest terms if the greatest common divisor (GCD) of a and b is 111, meaning the numbers are coprime or relatively prime.

Fractions in lowest terms can also extend to algebraic expressions. When dealing with algebraic fractions, the process involves factorizing both the numerator and denominator polynomials, then canceling common factors. For instance, to simplify an algebraic fraction like x2−5x+6x2−9\frac{x^2 - 5x + 6}{x^2 - 9}x2−9x2−5x+6​, we factorize the polynomials in both parts and cancel any common factors to obtain the fraction in its lowest form.

Examples of Reducing Fractions to Lowest Terms Example 1: Reducing a Fraction Through Common Factors Problem:

Reduce 4860\frac{48}{60}6048​ to lowest terms.

Step-by-step solution:

Step 1, Look for a common factor of both numbers. Since both 48 and 60 are even, we can divide by 2: 48÷260÷2=2430\frac{48 \div 2}{60 \div 2} = \frac{24}{30}60÷248÷2​=3024​

Step 2, The fraction 2430\frac{24}{30}3024​ still has common factors. Divide by 222 again: 24÷230÷2=1215\frac{24 \div 2}{30 \div 2} = \frac{12}{15}30÷224÷2​=1512​

Step 3, The fraction 1215\frac{12}{15}1512​ still has a common factor of 3: 12÷315÷3=45\frac{12 \div 3}{15 \div 3} = \frac{4}{5}15÷312÷3​=54​

Step 4, Now check if 444 and 555 have any common factors. Since they don't, 45\frac{4}{5}54​ is our final answer.

Second approach: Using the Greatest Common Divisor (GCD)

Step 1: Find the prime factorizations: 48=24×3=2×2×2×2×348 = 2^4 \times 3 = 2 \times 2 \times 2 \times 2 \times 348=24×3=2×2×2×2×3 60=22×3×5=2×2×3×560 = 2^2 \times 3 \times 5 = 2 \times 2 \times 3 \times 560=22×3×5=2×2×3×5

Step 2: Identify common prime factors: 22×3=122^2 \times 3 = 1222×3=12

Step 3: Divide both numbers by their GCD: 48÷1260÷12=45\frac{48 \div 12}{60 \div 12} = \frac{4}{5}60÷1248÷12​=54​

Therefore, 4860\frac{48}{60}6048​ in lowest terms is 45\frac{4}{5}54​.

Example 2: Simplifying a Fraction When the GCD Equals the Numerator Problem:

Express 105945\frac{105}{945}945105​ in lowest terms.

Step-by-step solution:

Step 1, Find the prime factorization of both numbers: 105=3×5×7105 = 3 \times 5 \times 7105=3×5×7 945=33×5×7=3×3×3×5×7945 = 3^3 \times 5 \times 7 = 3 \times 3 \times 3 \times 5 \times 7945=33×5×7=3×3×3×5×7

Step 2, Identify the greatest common divisor (GCD) by finding all shared prime factors: GCD(105,945)=3×5×7=105\text{GCD}(105, 945) = 3 \times 5 \times 7 = 105GCD(105,945)=3×5×7=105

Note: Notice that all factors of 105 appear in 945, making 105 itself the GCD.

Step 3, Divide both the numerator and denominator by the GCD: 105÷105945÷105=19\frac{105 \div 105}{945 \div 105} = \frac{1}{9}945÷105105÷105​=91​

Therefore, 105945\frac{105}{945}945105​ in lowest terms is 19\frac{1}{9}91​.

Example 3: Reducing a Fraction with Multiple Prime Factors Problem:

Express 126210\frac{126}{210}210126​ in lowest terms.

Step-by-step solution:

Step 1, Find the prime factorization of both numbers: 126=2×32×7126 = 2 \times 3^2 \times 7126=2×32×7 210=2×3×5×7210 = 2 \times 3 \times 5 \times 7210=2×3×5×7

Step 2, Identify the greatest common divisor (GCD) by finding all shared prime factors: GCD(126,210)=2×3×7=42\text{GCD}(126, 210) = 2 \times 3 \times 7 = 42GCD(126,210)=2×3×7=42

Step 3, Divide both the numerator and denominator by the GCD: 126÷42210÷42=35\frac{126 \div 42}{210 \div 42} = \frac{3}{5}210÷42126÷42​=53​

Therefore, 126210\frac{126}{210}210126​ in lowest terms is 35\frac{3}{5}53​.

Comments(7)AActorQuinnNovember 4, 2025I've used this lowest terms definition with my students. It's clear and the examples helped them grasp simplifying fractions quickly!

NNatureLover75September 17, 2025This glossary page was super helpful! I used the Lowest Terms examples to explain fractions to my kids, and they finally got it. The step-by-step breakdown makes it so much easier to teach!

NNatureLover75September 10, 2025I’ve been trying to help my kids simplify fractions, and this page was a lifesaver! The clear examples on reducing fractions to lowest terms made it so much easier for them to understand. Thanks for breaking it down so well!

MCMs. CarterAugust 27, 2025I used the Lowest Terms definition and examples from this page to help my kids with their math homework. The step-by-step explanations made it so easy for them to understand simplifying fractions. Thanks for breaking it down so clearly!

MCMs. CarterAugust 20, 2025This explanation of lowest terms was so clear! I used it to help my kids simplify their fractions, and the step-by-step examples really made it click for them. Thanks for making math less intimidating!

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