Simplifying with a Small Common Factor

Simplify $\frac{6}{10}$

Find a common factor: Both 6 and 10 are divisible by 2 = Common factor: 2

Divide numerator and denominator: $\frac{6 \div 2}{10 \div 2} = \frac{3}{5}$ = $\frac{3}{5}$

Check if simplified: 3 and 5 share no common factors (except 1) = In simplest form

Answer: $\frac{6}{10} = \frac{3}{5}$

Simplifying in Multiple Steps

Simplify $\frac{24}{36}$

Find a common factor: Both are even, so divisible by 2 = Divide by 2

First simplification: $\frac{24 \div 2}{36 \div 2} = \frac{12}{18}$ = $\frac{12}{18}$

Still have common factors?: 12 and 18 are both divisible by 2 = Divide by 2 again

Second simplification: $\frac{12 \div 2}{18 \div 2} = \frac{6}{9}$ = $\frac{6}{9}$

Check again: 6 and 9 are both divisible by 3 = Divide by 3

Final simplification: $\frac{6 \div 3}{9 \div 3} = \frac{2}{3}$ = $\frac{2}{3}$ - simplest form!

Answer: $\frac{24}{36} = \frac{2}{3}$

Using the Greatest Common Factor (GCF)

Simplify $\frac{18}{24}$ using the GCF method

List factors of 18: 1, 2, 3, 6, 9, 18 = Factors of 18

List factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 = Factors of 24

Find the GCF: Common factors: 1, 2, 3, 6. Largest is 6 = GCF = 6

Divide both by GCF: $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$ = $\frac{3}{4}$

Answer: $\frac{18}{24} = \frac{3}{4}$ (using GCF of 6)

Already in Simplest Form

Simplify $\frac{7}{12}$

List factors of 7: 1, 7 (7 is prime!) = Only factors: 1 and 7

Check if 12 is divisible by 7: $12 \div 7 = 1.71...$ (not a whole number) = 7 does not divide 12

Conclusion: The only common factor is 1 = Already simplified!

Answer: $\frac{7}{12}$ is already in simplest form

Mistake: Subtracting instead of dividing: $\frac{6}{8} \rightarrow \frac{6-2}{8-2} = \frac{4}{6}$

Why: Subtracting the same number changes the fraction's value. Only dividing keeps it equivalent.

Correct: Always DIVIDE both parts by the same number: $\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

Mistake: Dividing only the numerator or only the denominator

Why: You must do the same operation to both parts to maintain the fraction's value.

Correct: Whatever you do to the top, do to the bottom: $\frac{10 \div 5}{15 \div 5} = \frac{2}{3}$

Mistake: Stopping before reaching simplest form: $\frac{12}{16} = \frac{6}{8}$ and stopping

Why: $\frac{6}{8}$ still has a common factor of 2.

Correct: Keep going! $\frac{6}{8} = \frac{3}{4}$ - now it's fully simplified

Mistake: Thinking you can only divide by 2

Why: Any common factor works! Sometimes 3, 5, or larger numbers are the GCF.

Correct: For $\frac{15}{25}$, divide by 5: $\frac{15 \div 5}{25 \div 5} = \frac{3}{5}$

Cooking and Recipes

Recipes are written with simplified fractions for clarity.

A recipe calls for $\frac{4}{8}$ cup of sugar. Simplified, that's $\frac{1}{2}$ cup - much easier to measure!

Sharing Equally

When dividing things fairly, simplified fractions show the clearest portions.

If 12 out of 16 students finished their homework, that's $\frac{12}{16} = \frac{3}{4}$ of the class.

Discounts and Sales

Stores often express discounts as simplified fractions.

A "20 out of 100" discount is $\frac{20}{100} = \frac{1}{5}$ off, or "one-fifth off."

Q: How do I know when a fraction is fully simplified?

A: A fraction is fully simplified when the numerator and denominator have no common factors except 1. Check: can any number (2, 3, 5, etc.) divide both evenly? If not, you're done!

Q: Does simplifying change the value of a fraction?

A: No! Simplifying creates an equivalent fraction. $\frac{6}{8}$ and $\frac{3}{4}$ represent the exact same amount - just written differently.

Q: What if the numerator is larger than the denominator?

A: You can still simplify! For example, $\frac{10}{4} = \frac{5}{2}$. This is an improper fraction, which can also be written as the mixed number $2\frac{1}{2}$.