Operations with Fractions

🍕 Operations with Fractions

Part 1 of 5 — Fraction Foundations

Topics in This Part

🔑 Key Concept: A fraction is just a division problem in disguise — $\dfrac{a}{b}$ means "$a$ pieces, each one $b$-th of a whole." Master what fractions mean in Part 1 and every operation later becomes common sense.

What a Fraction Really Means

Every fraction has two parts:

$\frac{\text{numerator}}{\text{denominator}} = \frac{3}{4}$

• The denominator ($4$) tells you how many equal pieces the whole is split into.
• The numerator ($3$) tells you how many of those pieces you have.

So $\dfrac{3}{4}$ of a pizza means the pizza was cut into $4$ equal slices and you took $3$ of them.

Three Kinds of Fractions

💡 $\dfrac{7}{4}$ and $1\tfrac{3}{4}$ are the same amount — one written as a single fraction, one split into a whole plus a part.

Concept Check 🎯

Equivalent Fractions

Two fractions are equivalent when they name the same amount. You build one by multiplying (or dividing) the top and bottom by the same nonzero number:

$\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}, \qquad \frac{1}{2} = \frac{1 \cdot 5}{2 \cdot 5} = \frac{5}{10}$

Half a pizza is the same whether you cut it into $2$, $6$, or $10$ slices.

⚠️ You may only multiply or divide top and bottom by the same number. Adding the same number to both does not give an equivalent fraction — $\dfrac{1}{2} \ne \dfrac{1+1}{2+1} = \dfrac{2}{3}$.

Build Equivalent Fractions 🧮

Fill in the missing numerator.

1) $\dfrac{2}{3} = \dfrac{?}{12}$ 2) $\dfrac{3}{5} = \dfrac{?}{20}$ 3) $\dfrac{1}{4} = \dfrac{?}{16}$

Simplifying to Lowest Terms

A fraction is in lowest terms when the numerator and denominator share no common factor except $1$. To simplify, divide top and bottom by their greatest common factor (GCF).

Example: Simplify $\dfrac{12}{18}$

The GCF of $12$ and $18$ is $6$:

$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

Example: Simplify $\dfrac{15}{20}$

The GCF of $15$ and $20$ is $5$:

$\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

🔑 Always give your final answer in lowest terms — it is the standard way to report a fraction, and it makes answers easy to compare.

Simplify Each Fraction 🔽

Choose the lowest-terms form of each fraction.

🍕 Operations with Fractions

Part 2 of 5 — Adding & Subtracting Fractions

🔑 The Golden Rule: You can only add or subtract fractions when they have the same denominator. Same-size pieces add directly; different-size pieces must be rewritten first.

Like Denominators (the easy case)

When denominators match, add or subtract the numerators and keep the denominator the same:

$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$

Examples

$\frac{2}{7} + \frac{3}{7} = \frac{5}{7} \qquad \frac{5}{8} - \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$

⚠️ Never add the denominators! $\dfrac{2}{7} + \dfrac{3}{7}$ is $\dfrac{5}{7}$, not $\dfrac{5}{14}$. The denominator names the size of the pieces, and the pieces don't change when you combine them.

Always simplify at the end: $\dfrac{4}{8}$ becomes $\dfrac{1}{2}$.

Like Denominators 🧮

Add or subtract. Give each answer in lowest terms. Enter fractions like 3/5.

1) $\dfrac{1}{6} + \dfrac{4}{6} = \,?$ 2) $\dfrac{7}{10} - \dfrac{1}{10} = \,?$ 3) $\dfrac{3}{8} + \dfrac{1}{8} = \,?$

Unlike Denominators

Different denominators mean different-size pieces, so first rewrite both fractions with a common denominator — usually the least common multiple (LCM) of the two denominators, called the LCD.

Example: $\dfrac{1}{3} + \dfrac{1}{4}$

The LCD of $3$ and $4$ is $12$. Rewrite each fraction over $12$:

$\frac{1}{3} = \frac{4}{12}, \qquad \frac{1}{4} = \frac{3}{12}$

Now the denominators match, so add the numerators:

$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Example: $\dfrac{5}{6} - \dfrac{1}{4}$

The LCD of $6$ and $4$ is $12$:

$\frac{5}{6} = \frac{10}{12}, \qquad \frac{1}{4} = \frac{3}{12} \;\Rightarrow\; \frac{10}{12} - \frac{3}{12} = \frac{7}{12}$

💡 Finding the LCD: list multiples of the larger denominator until one is also a multiple of the smaller. Multiples of $6$: $6, 12$ — and $12$ is divisible by $4$, so $\text{LCD} = 12$.

Concept Check 🎯

Unlike Denominators 🧮

Find a common denominator, add or subtract, then simplify. Enter fractions like 7/12.

1) $\dfrac{1}{2} + \dfrac{1}{3} = \,?$ 2) $\dfrac{3}{4} - \dfrac{1}{6} = \,?$ 3) $\dfrac{2}{3} + \dfrac{1}{6} = \,?$

🍕 Operations with Fractions

Part 3 of 5 — Multiplying & Dividing Fractions

🔑 Good news: Multiplying and dividing fractions is actually easier than adding them — you do not need a common denominator. Just follow two short rules.

Multiplying Fractions

Multiply straight across — numerator times numerator, denominator times denominator:

$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$

Example

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

Cancel Before You Multiply

You can simplify before multiplying by canceling a common factor from any top with any bottom. It keeps the numbers small:

$\frac{3}{4} \cdot \frac{8}{9} = \frac{\cancel{3}^{\,1}}{\cancel{4}^{\,1}} \cdot \frac{\cancel{8}^{\,2}}{\cancel{9}^{\,3}} = \frac{1 \cdot 2}{1 \cdot 3} = \frac{2}{3}$

Here $4$ and $8$ share a factor of $4$ (leaving $1$ and $2$), and $3$ and $9$ share a factor of $3$ (leaving $1$ and $3$).

💡 "Of" means multiply: $\dfrac{1}{2}$ of $\dfrac{3}{5}$ is $\dfrac{1}{2} \cdot \dfrac{3}{5} = \dfrac{3}{10}$.

Multiply Fractions 🧮

Multiply and write each answer in lowest terms. Enter fractions like 3/10.

1) $\dfrac{2}{5} \cdot \dfrac{1}{3} = \,?$ 2) $\dfrac{3}{4} \cdot \dfrac{2}{9} = \,?$ 3) $\dfrac{5}{6} \cdot \dfrac{3}{10} = \,?$

Dividing Fractions

To divide, multiply by the reciprocal of the second fraction (flip it):

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

The reciprocal of $\dfrac{c}{d}$ is $\dfrac{d}{c}$ — just swap top and bottom.

Example: $\dfrac{1}{2} \div \dfrac{3}{4}$

Flip the second fraction and multiply:

$\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \cdot \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$

🔑 Many students remember this as "Keep, Change, Flip": keep the first fraction, change $\div$ to $\times$, flip the second fraction.

⚠️ Only flip the second fraction (the divisor). Flipping the first one gives the wrong answer.

Keep, Change, Flip 🔽

You are computing $\dfrac{2}{3} \div \dfrac{5}{6}$. Choose what happens at each step.

Divide Fractions 🧮

Use Keep, Change, Flip. Give answers in lowest terms. Enter fractions like 4/9.

1) $\dfrac{1}{3} \div \dfrac{2}{5} = \,?$ 2) $\dfrac{4}{5} \div \dfrac{2}{3} = \,?$ 3) $\dfrac{3}{8} \div \dfrac{3}{4} = \,?$

🍕 Operations with Fractions

Part 4 of 5 — Mixed Numbers & Word Problems

🔑 Big idea: To do any operation on a mixed number, first turn it into an improper fraction. Then it behaves exactly like the fractions you already know.

Converting Mixed Numbers

Mixed number → improper fraction: multiply the whole number by the denominator, add the numerator, keep the denominator.

$2\frac{3}{4} = \frac{(2 \cdot 4) + 3}{4} = \frac{11}{4}$

Improper fraction → mixed number: divide. The quotient is the whole number, the remainder is the new numerator.

$\frac{17}{5} = 17 \div 5 = 3 \text{ R } 2 = 3\frac{2}{5}$

💡 Check $2\tfrac{5}{6}$: $(2 \cdot 6) + 5 = 17$, so $\dfrac{17}{6}$. ✓

Convert to Improper Fractions 🧮

Rewrite each mixed number as an improper fraction. Enter like 11/4.

1) $3\dfrac{1}{2} = \,?$ 2) $2\dfrac{2}{3} = \,?$ 3) $4\dfrac{1}{5} = \,?$

Operating with Mixed Numbers

Example: $1\dfrac{1}{2} \cdot 2\dfrac{2}{3}$

Convert both, then multiply:

$1\frac{1}{2} \cdot 2\frac{2}{3} = \frac{3}{2} \cdot \frac{8}{3} = \frac{24}{6} = 4$

Example: $3\dfrac{1}{4} - 1\dfrac{1}{2}$

Convert both to a common denominator of $4$:

$3\frac{1}{4} - 1\frac{1}{2} = \frac{13}{4} - \frac{6}{4} = \frac{7}{4} = 1\frac{3}{4}$

⚠️ When the final answer is improper, convert it back to a mixed number unless the problem asks otherwise — $\dfrac{7}{4} = 1\tfrac{3}{4}$.

Word Problems 🎯

Mixed Number Practice 🧮

Solve. Enter improper-fraction answers in lowest terms like 7/4, or a whole number like 4.

1) $1\dfrac{1}{3} + 2\dfrac{1}{3} = \,?$ (give as improper, lowest terms) 2) $2\dfrac{1}{2} \cdot \dfrac{2}{5} = \,?$ (whole number) 3) $3\dfrac{1}{2} \div 1\dfrac{3}{4} = \,?$ (whole number)

🍕 Operations with Fractions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) simplify fractions, (2) add and subtract with common denominators, (3) multiply and divide, and (4) handle mixed numbers and word problems. Let's pull it all together.

Quick Reference

⚠️ Three habits that prevent most mistakes:

1. Find a common denominator only for $+$ and $-$.
2. Flip the second fraction when dividing.
3. Always give the final answer in lowest terms.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.