To multiply two rational numbers, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
Basic Multiplication: $-\frac{5}{3} \times \frac{7}{4} = -\frac{35}{12}$
Simplifying Before Multiplying: You can save time by “canceling out” factors. For example, in $-\frac{5}{3} \times \frac{9}{4}$, the $3$ in the denominator and the $9$ in the numerator can both be divided by $3$, leaving you with $-\frac{15}{4}$.
Whole Numbers: Treat a whole number like it has a denominator of $1$. So, $-6 \times \frac{24}{25} = -\frac{144}{25}$.
The Zero Property: Anything multiplied by zero is always zero. $0 \times \frac{1}{2} = 0$.
When you have a long string of numbers to multiply, look for common factors across all numerators and denominators.
Example Challenge:
By identifying that $37$ goes into $111$ exactly $3$ times, and simplifying other factors like $35$ and $56$ (both divisible by $7$), a complex problem becomes much simpler. Following the cancellations shown in the worksheet, the final result is:
Note on Signs: Remember that a negative times a negative results in a positive answer!
Before we can divide, we have to understand the Reciprocal. To find the reciprocal of a fraction, simply flip it upside down.
The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.
Division is just “multiplication in disguise.” To divide, you keep the first fraction exactly as it is, change the division sign to multiplication, and flip the second fraction (use its reciprocal).
Check the signs: Count your negatives (even amount = positive result; odd amount = negative).
Simplify first: Cross-cancel numbers to keep the math easy.
Multiply across: Top $\times$ Top and Bottom $\times$ Bottom.
Flip for Division: Always use the reciprocal of the second number.
When you see one fraction divided by another, you multiply the top fraction by the flipped version of the bottom fraction:
If you are dividing a fraction by a whole number, remember that a whole number $c$ is the same as $\frac{c}{1}$. Therefore, its reciprocal is $\frac{1}{c}$.
1. Basic Division with Simplification
Calculate: $\frac{2}{3} \div \frac{5}{6}$
Step 1: Multiply the first fraction by the reciprocal of the second: $\frac{2}{3} \times \frac{6}{5}$.
Step 2: Simplify by cross-canceling (the $3$ and $6$ reduce to $1$ and $2$): $\frac{2}{1} \times \frac{2}{5}$.
Result: $\mathbf{\frac{4}{5}}$.
2. Fraction Divided by a Whole Number
Calculate: $\frac{2}{3} \div 2$
Step 1: Rewrite as multiplication by the reciprocal of $2$ (which is $\frac{1}{2}$): $\frac{2}{3} \times \frac{1}{2}$.
Step 2: The $2$ in the numerator and denominator cancel out.
Result: $\mathbf{\frac{1}{3}}$.
3. Dividing Mixed Numbers
Calculate: Divide $1\frac{2}{3}$ by $7\frac{8}{9}$
Step 1: Convert to improper fractions. $1\frac{2}{3}$ becomes $\frac{5}{3}$ and $7\frac{8}{9}$ becomes $\frac{71}{9}$.
Step 2: Set up the multiplication. $\frac{5}{3} \times \frac{9}{71}$.
Step 3: Simplify. Cross-cancel the $3$ and $9$ to get $1$ and $3$.
Step 4: Multiply. $5 \times 3 = 15$ and $1 \times 71 = 71$.
Result: $\mathbf{\frac{15}{71}}$.