1.5 Visualize Fractions

Fractions

A fraction represents a part-to-whole relationship. The top number (numerator) tells you how many parts you have, and the bottom number (denominator) tells you how many equal parts make up the whole. Fractions show up constantly in algebra, so getting comfortable with them now will pay off throughout the course.

Equivalent Fractions

Two fractions are equivalent when they represent the same value, even though they use different numerators and denominators. You create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.

For example, to find a fraction equivalent to $\frac{3}{5}$, multiply both parts by 2:

$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$

Both $\frac{3}{5}$ and $\frac{6}{10}$ land on the same spot on a number line. This works because multiplying top and bottom by the same number is really just multiplying by 1 (since $\frac{2}{2} = 1$), which doesn't change the value.

• $\frac{2}{3} = \frac{4}{6}$ because $2 \times 2 = 4$ and $3 \times 2 = 6$
• $\frac{2}{4} = \frac{1}{2}$ because $2 \div 2 = 1$ and $4 \div 2 = 2$
• Equivalent fractions are especially useful when you need to compare fractions that have different denominators

Simplifying Fractions

A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factor other than 1. To simplify, divide both the numerator and denominator by their greatest common factor (GCF).

Steps to simplify a fraction:

1. Find the GCF of the numerator and denominator.
2. Divide both the numerator and denominator by that GCF.
3. Check that no common factor remains.

For example, simplify $\frac{6}{8}$:

• The GCF of 6 and 8 is 2.
• $\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

The fractions $\frac{6}{8}$ and $\frac{3}{4}$ are equal, but $\frac{3}{4}$ is easier to read and work with. Always simplify your final answer after multiplying or dividing fractions.

Multiplication and Division of Fractions

Multiplying fractions: Multiply the numerators together and multiply the denominators together.

$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

Dividing fractions: Multiply the first fraction by the reciprocal of the second. The reciprocal of a fraction is what you get when you flip the numerator and denominator. So the reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$.

$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

A common way to remember division: "Keep, Change, Flip." Keep the first fraction, change division to multiplication, and flip the second fraction.

• Always simplify your result to lowest terms.
• You can also simplify before multiplying by canceling common factors between any numerator and any denominator. This keeps the numbers smaller and saves a step at the end.

Fraction Bars in Expressions

A fraction bar acts as a grouping symbol, just like parentheses. It tells you to simplify everything in the numerator and everything in the denominator separately, then divide.

$\frac{a + b}{c} \text{ means } (a + b) \div c$

For example:

$\frac{2 + 4}{3 + 1} = \frac{6}{4} = \frac{3}{2}$

You must finish all the operations on top and all the operations on bottom before you divide. Treat the numerator and denominator as if they each have invisible parentheses around them.

Verbal to Fractional Conversions

Word problems often describe fractions without using fraction notation. Key phrases to watch for:

• "out of" → "3 out of 5" means $\frac{3}{5}$
• "divided by" → "7 divided by 10" means $\frac{7}{10}$
• "the ratio of __ to __" → "the ratio of a to b" means $\frac{a}{b}$
• "per" → can also signal division

Pay attention to word order. The number mentioned first typically becomes the numerator, and the number after "out of," "to," or "divided by" becomes the denominator.

Fractions and Their Representations

Fractions express a part-whole relationship where the numerator is divided by the denominator. Every fraction can also be written in other forms:

• Decimal: Divide the numerator by the denominator. For example, $\frac{1}{2} = 0.5$.
• Percentage: Multiply the decimal form by 100. For example, $\frac{1}{2} = 0.5 = 50\%$.

All three forms (fraction, decimal, percentage) represent the same value. Being able to move between them is a skill you'll use often, especially when comparing quantities or interpreting real-world data.