Finding an Equivalent Fraction by Multiplying

Find a fraction equivalent to $\frac{2}{3}$ with a denominator of 12.

Identify what to multiply the denominator by: $3 \times ? = 12$, so $3 \times 4 = 12$ = Multiply by 4

Multiply both numerator and denominator by the same number: $\frac{2}{3} \times \frac{4}{4} = \frac{2 \times 4}{3 \times 4}$ = $\frac{8}{12}$

Verify the fractions are equivalent: $\frac{2}{3} = \frac{8}{12}$ (both equal approximately 0.667) = Confirmed equivalent

Answer: $\frac{2}{3} = \frac{8}{12}$

Simplifying a Fraction to Find an Equivalent

Simplify $\frac{6}{8}$ to its lowest terms.

Find the greatest common factor (GCF) of 6 and 8: Factors of 6: 1, 2, 3, 6. Factors of 8: 1, 2, 4, 8. GCF = 2 = GCF is 2

Divide both numerator and denominator by the GCF: $\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$ = $\frac{3}{4}$

Check if it can be simplified further: 3 and 4 share no common factors except 1 = Already in lowest terms

Answer: $\frac{6}{8} = \frac{3}{4}$

Checking if Two Fractions are Equivalent

Are $\frac{3}{5}$ and $\frac{9}{15}$ equivalent fractions?

Method 1: Cross multiply: $3 \times 15 = 45$ and $5 \times 9 = 45$ = Both products equal 45

Method 2: Simplify $\frac{9}{15}$: $\frac{9 \div 3}{15 \div 3} = \frac{3}{5}$ = Simplifies to $\frac{3}{5}$

Compare the results: Both methods confirm $\frac{3}{5} = \frac{9}{15}$ = Yes, they are equivalent

Answer: Yes, $\frac{3}{5} = \frac{9}{15}$

Mistake: Adding the same number to both numerator and denominator

Why: $\frac{1}{2} + \frac{1}{1} \neq \frac{2}{3}$. Adding doesn't preserve the ratio between parts and whole.

Correct: Always multiply or divide both by the same number: $\frac{1}{2} \times \frac{2}{2} = \frac{2}{4}$

Mistake: Only changing the numerator or only the denominator

Why: Changing just one number changes the value of the fraction entirely.

Correct: Both numerator AND denominator must be multiplied (or divided) by the same number.

Mistake: Thinking larger numbers always mean larger fractions

Why: $\frac{6}{12}$ looks bigger than $\frac{1}{2}$ but they're equal! The relationship between numerator and denominator matters.

Correct: Compare by finding common denominators or simplifying first.

Cooking and Recipes

Recipes often need to be scaled up or down, requiring equivalent fractions.

If a recipe calls for $\frac{3}{4}$ cup of flour and you want to double it, you need $\frac{6}{4}$ or $\frac{3}{2}$ cups.

Pizza and Fair Sharing

Understanding that different slices can represent equal amounts helps with fair sharing.

A pizza cut into 8 slices: 4 slices ($\frac{4}{8}$) equals half the pizza ($\frac{1}{2}$).

Money and Coins

Coins represent fractional parts of a dollar in equivalent ways.

Half a dollar ($\frac{1}{2}$) = 2 quarters ($\frac{2}{4}$) = 5 dimes ($\frac{5}{10}$) = 50 cents ($\frac{50}{100}$)

Q: How do I know if a fraction is in lowest terms?

A: A fraction is in lowest terms when the numerator and denominator share no common factors except 1. For example, $\frac{3}{4}$ is in lowest terms, but $\frac{6}{8}$ is not (both divisible by 2).

Q: Can I find infinite equivalent fractions?

A: Yes! You can multiply the numerator and denominator by any number: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{100}{200}$ and so on forever.

Q: Why does multiplying by $\frac{2}{2}$ not change the fraction's value?

A: Because $\frac{2}{2} = 1$, and multiplying by 1 never changes a number's value. It just changes how the number looks.