Lesson 1.5: Visualize Fractions

New Concept

This lesson introduces fractions as parts of a whole, represented as $\frac{a}{b}$. You'll learn how to find equivalent fractions, simplify them, and master fundamental operations like multiplication and division to solve a variety of problems.

What’s next

Soon, you'll work through interactive examples and practice cards to master simplifying, multiplying, and dividing fractions.

Property

Equivalent fractions are fractions that have the same value.

Equivalent Fractions Property
If $a$, $b$, $c$ are numbers where $b \neq 0$, $c \neq 0$, then

Examples

• To find three fractions equivalent to $\frac{3}{4}$, we can multiply the numerator and denominator by 2, 3, and 5. This gives us $\frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8}$, $\frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12}$, and $\frac{3 \cdot 5}{4 \cdot 5} = \frac{15}{20}$.
• To find a fraction equivalent to $\frac{2}{7}$ with a denominator of 21, we see that $7 \cdot 3 = 21$. So we multiply the numerator by 3 as well: $\frac{2 \cdot 3}{7 \cdot 3} = \frac{6}{21}$.
• The fractions $\frac{5}{10}$, $\frac{10}{20}$, and $\frac{50}{100}$ are all equivalent to $\frac{1}{2}$ because they represent the same value.

Property

A fraction is considered simplified if there are no common factors in its numerator and denominator.

How to Simplify a Fraction

1. Rewrite the numerator and denominator to show the common factors. If needed, factor them into prime numbers first.
2. Simplify using the equivalent fractions property by dividing out common factors.
3. Multiply the remaining factors, if necessary.

Examples

• To simplify $-\frac{24}{40}$, we find the common factor of 8. We rewrite it as $-\frac{3 \cdot 8}{5 \cdot 8}$. Dividing out the 8 gives us $-\frac{3}{5}$.
• To simplify $\frac{150}{225}$, we can see a common factor of 25. $\frac{6 \cdot 25}{9 \cdot 25} = \frac{6}{9}$. This can be simplified further by a factor of 3: $\frac{2 \cdot 3}{3 \cdot 3} = \frac{2}{3}$.
• To simplify $\frac{9x}{9y}$, we can divide out the common factor 9. This leaves us with $\frac{x}{y}$.

Property

Fraction Multiplication
If $a, b, c, d$ are numbers where $b \neq 0$, $d \neq 0$, then

Examples

• To multiply $\frac{2}{5} \cdot \frac{3}{7}$, we multiply the numerators ($2 \cdot 3 = 6$) and the denominators ($5 \cdot 7 = 35$). The result is $\frac{6}{35}$.
• To multiply $-\frac{8}{9} \cdot \frac{3}{4}$, first determine the sign (negative). Then multiply: $\frac{8 \cdot 3}{9 \cdot 4} = \frac{24}{36}$. This simplifies to $\frac{2}{3}$, so the final answer is $-\frac{2}{3}$.
• To multiply $-\frac{15}{4}(-12y)$, first write $-12y$ as a fraction: $\frac{-12y}{1}$. The product is positive. $\frac{15 \cdot 12y}{4 \cdot 1} = \frac{15 \cdot 3 \cdot 4y}{4} = 45y$.

Explanation

Multiplying fractions is a direct process. You multiply the top numbers (numerators) together to get the new top number, and multiply the bottom numbers (denominators) together for the new bottom number. Then, simplify if needed.

Property

Reciprocal
The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$.

Fraction Division
If $a, b, c, d$ are numbers where $b \neq 0, c \neq 0, d \neq 0$, then

Examples

• To divide $\frac{2}{5} \div \frac{3}{4}$, we keep $\frac{2}{5}$, change $\div$ to $\cdot$, and flip $\frac{3}{4}$ to $\frac{4}{3}$. This gives $\frac{2}{5} \cdot \frac{4}{3} = \frac{8}{15}$.
• To divide $-\frac{5}{18} \div (-\frac{15}{24})$, the result is positive. We calculate $\frac{5}{18} \cdot \frac{24}{15} = \frac{5 \cdot 6 \cdot 4}{3 \cdot 6 \cdot 3 \cdot 5} = \frac{4}{9}$.
• To solve $\frac{5x}{7} \div \frac{10x}{21}$, we multiply by the reciprocal: $\frac{5x}{7} \cdot \frac{21}{10x}$. After canceling common factors, we get $\frac{1}{1} \cdot \frac{3}{2} = \frac{3}{2}$.

Property

A fraction bar acts as a grouping symbol. Use the order of operations to simplify the numerator and the denominator separately before simplifying the fraction as a whole.

Examples

• To simplify $\frac{7+3 \cdot 4}{2^3+1}$, first simplify the top to $7+12=19$ and the bottom to $8+1=9$. The result is $\frac{19}{9}$.
• To simplify $\frac{5-3(2)}{10-3}$, simplify the numerator to $5-6=-1$ and the denominator to $7$. The answer is $\frac{-1}{7}$ or $-\frac{1}{7}$.
• To simplify $\frac{(4-2)^2}{6+2}$, calculate the numerator as $2^2=4$ and the denominator as $8$. The fraction is $\frac{4}{8}$, which simplifies to $\frac{1}{2}$.

Explanation

Think of the fraction bar as a 'do this first' command for both the top and bottom expressions. Calculate the value of the numerator and the denominator independently. After that, perform the final division or simplify the fraction.

Property

The word quotient indicates division, which can be written as a fraction. The phrase 'the quotient of $A$ and $B$' is translated into the algebraic expression $\frac{A}{B}$. The dividend ($A$) becomes the numerator, and the divisor ($B$) becomes the denominator.

Examples

• The phrase 'the quotient of the sum of $a$ and $b$, and $c$' means you should first find the sum ($a+b$) and then divide by $c$. This is written as $\frac{a+b}{c}$.
• The phrase 'the quotient of $x$ and the difference of $y$ and 2' translates to the expression $\frac{x}{y-2}$.
• The phrase 'the quotient of the product of $5$ and $z$, and $w$' translates to the expression $\frac{5z}{w}$.

Explanation

When you read 'the quotient of' in a word problem, it's a signal to write a fraction. The first quantity mentioned after 'quotient of' goes on top (numerator), and the second quantity goes on the bottom (denominator).