Fractions
Master the art of working with parts of a whole
If fractions make you nervous, you are in very good company. Fractions trip up more people than almost any other topic in basic math. But here is the thing: you already work with fractions every single day. When you split a pizza with friends, when you notice your phone battery is at “half,” when you follow a recipe that calls for “three-quarters of a cup” - that is all fractions. You have been doing this your whole life. Now we are just going to give you the tools to work with fractions on paper as easily as you work with them in real life.
Core Concepts
What is a Fraction?
A fraction represents a part of a whole. When you cut a pizza into 8 equal slices and eat 3 of them, you have eaten $\frac{3}{8}$ of the pizza. That is a fraction.
Every fraction has two parts:
- Numerator (top number): How many parts you have
- Denominator (bottom number): How many equal parts the whole is divided into
Think of it this way: the denominator tells you the “size” of each piece, and the numerator tells you how many pieces you are dealing with.
$$\frac{\text{numerator}}{\text{denominator}} = \frac{\text{parts you have}}{\text{total equal parts}}$$
Types of Fractions
Proper Fractions have a numerator smaller than the denominator. These represent less than one whole: $$\frac{1}{2}, \quad \frac{3}{4}, \quad \frac{7}{8}$$
Improper Fractions have a numerator equal to or larger than the denominator. These represent one whole or more: $$\frac{5}{4}, \quad \frac{9}{3}, \quad \frac{11}{7}$$
Mixed Numbers combine a whole number with a proper fraction: $$1\frac{1}{2}, \quad 3\frac{3}{4}, \quad 2\frac{5}{8}$$
Do not let the names fool you - there is nothing wrong with improper fractions. In fact, they are often easier to work with in calculations. The name just means the fraction represents more than one whole.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Numerator | The top number in a fraction | In $\frac{3}{5}$, the numerator is 3 |
| Denominator | The bottom number in a fraction | In $\frac{3}{5}$, the denominator is 5 |
| Proper fraction | Numerator < denominator | $\frac{2}{7}$ |
| Improper fraction | Numerator ≥ denominator | $\frac{9}{4}$ |
| Mixed number | Whole number + fraction | $2\frac{1}{3}$ |
| Equivalent fractions | Different fractions with equal value | $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$ |
| Simplest form | Fraction reduced to lowest terms | $\frac{6}{8}$ simplifies to $\frac{3}{4}$ |
| Reciprocal | Fraction flipped upside down | Reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$ |
| Common denominator | Same denominator in multiple fractions | $\frac{2}{6}$ and $\frac{5}{6}$ share denominator 6 |
Examples
A recipe calls for $\frac{3}{4}$ cup of sugar. What does this mean?
Solution:
The denominator (4) tells us the cup is divided into 4 equal parts. The numerator (3) tells us we need 3 of those parts.
So $\frac{3}{4}$ cup means: divide a full cup into 4 equal parts, then use 3 of them.
If you do not have a $\frac{3}{4}$ measuring cup, you could use a $\frac{1}{4}$ cup three times - same result!
Convert $2\frac{3}{5}$ to an improper fraction, then convert $\frac{17}{4}$ to a mixed number.
Solution:
Mixed to improper: Multiply the whole number by the denominator, add the numerator, keep the same denominator.
$$2\frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$$
Think of it this way: 2 wholes is the same as $\frac{10}{5}$ (ten fifths), plus the extra $\frac{3}{5}$ gives you $\frac{13}{5}$.
Improper to mixed: Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator.
$$\frac{17}{4} = 17 \div 4 = 4 \text{ remainder } 1 = 4\frac{1}{4}$$
This makes sense: 17 quarters is 4 whole dollars and 1 quarter left over.
Calculate $\frac{2}{3} + \frac{3}{4}$.
Solution:
Before adding fractions, they must have the same denominator. We need to find a common denominator - a number that both 3 and 4 divide into evenly.
The smallest common denominator for 3 and 4 is 12 (since $3 \times 4 = 12$).
Convert each fraction to have denominator 12: $$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$$ $$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$$
Now add (just add the numerators, keep the denominator): $$\frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12}$$
Why this works: Imagine cutting pizzas. One pizza is cut into thirds, another into fourths. You cannot directly compare the slices because they are different sizes. But if you re-cut both pizzas into twelfths (same-size slices), now you can combine them.
You have $\frac{3}{4}$ of a pizza left. You eat $\frac{2}{3}$ of what is left. How much of the original pizza did you eat?
Solution:
To find a fraction of a fraction, multiply them.
$$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$$
You ate $\frac{1}{2}$ of the original pizza.
The rule: To multiply fractions, multiply numerators together and denominators together. No need for common denominators!
Tip: You can simplify before multiplying to make calculations easier. Notice the 3 in the numerator and denominator cancel: $$\frac{2}{\cancel{3}} \times \frac{\cancel{3}}{4} = \frac{2}{4} = \frac{1}{2}$$
A board is $3\frac{1}{2}$ feet long. You need pieces that are $\frac{2}{3}$ foot each. How many pieces can you cut?
Solution:
This is a division problem: $3\frac{1}{2} \div \frac{2}{3}$
Step 1: Convert the mixed number to an improper fraction. $$3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2}$$
Step 2: To divide fractions, multiply by the reciprocal (flip the second fraction and multiply). $$\frac{7}{2} \div \frac{2}{3} = \frac{7}{2} \times \frac{3}{2}$$
Step 3: Multiply. $$\frac{7}{2} \times \frac{3}{2} = \frac{21}{4} = 5\frac{1}{4}$$
You can cut 5 complete pieces, with a little left over (not enough for a sixth piece).
Why “flip and multiply” works: Division asks “how many of this fit into that?” When you flip and multiply, you are finding how many $\frac{2}{3}$-foot segments fit into $\frac{7}{2}$ feet.
Key Properties and Rules
Equivalent Fractions
Multiply or divide both numerator and denominator by the same number: $$\frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a \div n}{b \div n}$$
Simplifying Fractions
Divide both numerator and denominator by their greatest common factor (GCF): $$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$$
Comparing Fractions
To compare fractions, convert to the same denominator, then compare numerators: $$\frac{3}{4} \text{ vs } \frac{5}{6} \rightarrow \frac{9}{12} \text{ vs } \frac{10}{12} \rightarrow \frac{3}{4} < \frac{5}{6}$$
Adding and Subtracting Fractions
Same denominator: Add or subtract numerators, keep denominator. $$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$$
Different denominators: Find common denominator first, then add or subtract.
Multiplying Fractions
Multiply numerators, multiply denominators: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
Dividing Fractions
Multiply by the reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$$
Converting Mixed Numbers and Improper Fractions
Mixed to improper: $$a\frac{b}{c} = \frac{(a \times c) + b}{c}$$
Improper to mixed: Divide numerator by denominator; quotient is the whole number, remainder is the new numerator.
Real-World Applications
Cooking and Baking
Recipes are full of fractions. If a recipe serves 4 but you need to serve 6, you multiply all ingredients by $\frac{6}{4} = \frac{3}{2} = 1\frac{1}{2}$. That $\frac{3}{4}$ cup of flour becomes $\frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1\frac{1}{8}$ cups.
Money
A quarter is $\frac{1}{4}$ of a dollar. Three quarters is $\frac{3}{4}$ of a dollar, or 75 cents. If you have $\frac{2}{3}$ of your paycheck left after rent, and you spend $\frac{1}{4}$ of what remains on groceries, you spent $\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$ of your original paycheck on groceries.
Time
An hour has 60 minutes. Half an hour ($\frac{1}{2}$ hour) is 30 minutes. A quarter hour ($\frac{1}{4}$ hour) is 15 minutes. If a task takes $\frac{3}{4}$ hour and you have completed $\frac{2}{3}$ of it, you have spent $\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$ hour (30 minutes) so far.
Sports Statistics
A basketball player makes $\frac{7}{10}$ of free throws. In a season with 200 free throw attempts, they make $200 \times \frac{7}{10} = \frac{1400}{10} = 140$ shots.
Construction and DIY
Lumber and fabric are often measured in fractions. A board that is $5\frac{3}{4}$ inches wide cut down to $\frac{2}{3}$ its width becomes $\frac{23}{4} \times \frac{2}{3} = \frac{46}{12} = \frac{23}{6} = 3\frac{5}{6}$ inches wide.
Self-Test Problems
Problem 1: Simplify $\frac{24}{36}$ to its lowest terms.
Show Answer
Find the greatest common factor of 24 and 36, which is 12. $$\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}$$
Problem 2: Convert $4\frac{2}{7}$ to an improper fraction.
Show Answer
$$4\frac{2}{7} = \frac{(4 \times 7) + 2}{7} = \frac{28 + 2}{7} = \frac{30}{7}$$
Problem 3: Calculate $\frac{5}{6} - \frac{1}{4}$.
Show Answer
Find a common denominator (12): $$\frac{5}{6} = \frac{10}{12}, \quad \frac{1}{4} = \frac{3}{12}$$ $$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$$
Problem 4: A tank is $\frac{3}{5}$ full. If you use $\frac{1}{2}$ of what is in the tank, what fraction of the full tank have you used?
Show Answer
Multiply the fractions: $$\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}$$
You have used $\frac{3}{10}$ of the full tank.
Problem 5: Divide $2\frac{1}{3}$ by $\frac{7}{9}$.
Show Answer
Convert to improper fraction: $2\frac{1}{3} = \frac{7}{3}$
Flip and multiply: $$\frac{7}{3} \div \frac{7}{9} = \frac{7}{3} \times \frac{9}{7} = \frac{63}{21} = 3$$
Notice the 7s cancel: $\frac{\cancel{7}}{3} \times \frac{9}{\cancel{7}} = \frac{9}{3} = 3$
Summary
- A fraction represents parts of a whole, with the numerator (top) showing how many parts and the denominator (bottom) showing the total equal parts.
- Proper fractions ($\frac{3}{4}$) are less than 1; improper fractions ($\frac{5}{3}$) are 1 or greater; mixed numbers ($1\frac{2}{3}$) combine wholes and parts.
- Equivalent fractions have the same value: multiply or divide top and bottom by the same number.
- To add or subtract fractions, first get a common denominator, then work with the numerators.
- To multiply fractions, multiply straight across: numerators together, denominators together.
- To divide fractions, flip the second fraction and multiply (multiply by the reciprocal).
- Fractions appear everywhere in daily life: cooking, money, time, measurements, and more.
- When in doubt, draw a picture! Visualizing fractions as parts of a pie or bar can make operations much clearer.