Fractions, decimals, and percents for GED math

1. Equivalent fractions and simplifying fractions

You use equivalent fractions when you want a fraction in a different form but with the same value, and simplifying to make a fraction as simple and clear as possible.

What are equivalent fractions?

Two fractions are equivalent if they name the same part of a whole, even if the numbers look different.

Mechanism:

• If you multiply the numerator and denominator by the same nonzero number, the value doesn’t change.
• If you divide the numerator and denominator by the same nonzero number, the value doesn’t change.

Example:
$\dfrac{1}{2}$ and $\dfrac{3}{6}$

• Multiply $\dfrac{1}{2}$ top and bottom by 3:

$$\dfrac{1}{2} = \dfrac{1 \times 3}{2 \times 3} = \dfrac{3}{6}$$

They are equivalent because you’ve scaled both parts the same way.

How to simplify a fraction (reduce to lowest terms)

To simplify a fraction, you divide numerator and denominator by their greatest common factor (GCF).

Steps:

1. List a few factors of the numerator and denominator.
2. Find the largest factor they share.
3. Divide both top and bottom by that number.

Example 1: Simplify $\dfrac{18}{24}$

• Factors of 18: $1, 2, 3, 6, 9, 18$
• Factors of 24: $1, 2, 3, 4, 6, 8, 12, 24$
• Greatest common factor is 6.

Divide top and bottom by 6:

$$\dfrac{18}{24} = \dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}$$

So $\dfrac{18}{24}$ simplifies to $\dfrac{3}{4}$.

Example 2: Compare $\dfrac{2}{3}$ and $\dfrac{3}{5}$

To compare, you can:

• Turn them into like denominators (common denominator), or
• Turn them into decimals, or
• Cross-multiply.

Let’s cross-multiply:

• For $\dfrac{2}{3}$ and $\dfrac{3}{5}$, compare $2 \times 5$ and $3 \times 3$.
• $2 \times 5 = 10$
• $3 \times 3 = 9$

Because $10 > 9$, we know:

$$\dfrac{2}{3} > \dfrac{3}{5}$$

Visualizing equivalence (optional picture in your head)

Imagine a chocolate bar:

• Cut one bar into 2 equal pieces, take 1 → $\dfrac{1}{2}$
• Cut another bar into 6 equal pieces, take 3 → $\dfrac{3}{6}$

You’d see the same amount of chocolate shaded in both cases. That’s equivalence.