Before You Begin

Arithmetic operations — numerator and denominator calculations
Division — a fraction is division in another form

½ Fractions, Decimals & Percents

📚 Elementary & Middle School Math 🎯 Key Concepts: Simplifying, Operations, Conversions, Ratios

1. What Is a Fraction?

A fraction represents a part of a whole. It is written as \(\dfrac{a}{b}\), where a is the numerator (how many parts you have) and b is the denominator (how many equal parts the whole is divided into).

Types of Fractions

Proper fraction: numerator < denominator. Example: \(\dfrac{3}{5}\) (value less than 1)
Improper fraction: numerator ≥ denominator. Example: \(\dfrac{7}{4}\) (value ≥ 1)
Mixed number: a whole number plus a proper fraction. Example: \(1\dfrac{3}{4}\)

Intuition First

If you cut a pizza into 8 equal slices, each slice is 1/8. A fraction tells you how many parts you have out of a total number of equal parts.

Converting between improper fractions and mixed numbers:

Improper → Mixed: \(\dfrac{11}{4} = 2\dfrac{3}{4}\) (11 ÷ 4 = 2 remainder 3)

Mixed → Improper: \(2\dfrac{3}{4} = \dfrac{2 \times 4 + 3}{4} = \dfrac{11}{4}\)

2. Equivalent Fractions and Simplifying

Two fractions are equivalent if they represent the same value. You can multiply or divide both numerator and denominator by the same nonzero number.

\[ \frac{a}{b} = \frac{a \times k}{b \times k} \quad (k \neq 0) \]

Simplifying (Reducing) a Fraction

Divide both numerator and denominator by their Greatest Common Factor (GCF). A fraction is in simplest form (lowest terms) when the GCF of numerator and denominator is 1.

Simplify \(\dfrac{18}{24}\):

GCF(18, 24) = 6
\(\dfrac{18}{24} = \dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}\)

3. Adding and Subtracting Fractions

Same Denominator

\[ \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}, \qquad \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} \]

\(\dfrac{5}{8} + \dfrac{1}{8} = \dfrac{6}{8} = \dfrac{3}{4}\)

Different Denominators — Find the LCD

Find the Least Common Denominator (LCD) — the LCM of the two denominators — then convert each fraction before adding or subtracting.

Calculate \(\dfrac{1}{3} + \dfrac{3}{4}\):

LCD(3, 4) = 12
\(\dfrac{1}{3} = \dfrac{4}{12}\), \(\dfrac{3}{4} = \dfrac{9}{12}\)
\(\dfrac{4}{12} + \dfrac{9}{12} = \dfrac{13}{12} = 1\dfrac{1}{12}\)

💡 Why it works

You can't add 1/2 + 1/3 directly because the pieces are different sizes. Finding a common denominator means cutting everything into the same-size pieces first.

4. Multiplying and Dividing Fractions

Multiplication

\[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]

Tip — Cross-Cancel: Before multiplying, cancel any factor that appears in a numerator and a different denominator. This keeps numbers small.

\(\dfrac{4}{9} \times \dfrac{3}{8} = \dfrac{4 \times 3}{9 \times 8}\) → cancel 4/8 and 3/9 → \(= \dfrac{1}{6}\)

Division — "Keep, Change, Flip"

\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]

\(\dfrac{3}{5} \div \dfrac{2}{7} = \dfrac{3}{5} \times \dfrac{7}{2} = \dfrac{21}{10} = 2\dfrac{1}{10}\)

5. Decimals

Decimals are another way to write fractions whose denominators are powers of 10.

Place Value for Decimals

Tens	Ones	.	Tenths	Hundredths	Thousandths
10	1	.	0.1	0.01	0.001

Converting Fractions to Decimals

Divide the numerator by the denominator.

\(\dfrac{3}{4} = 3 \div 4 = 0.75\) \(\dfrac{1}{3} = 0.\overline{3}\) (repeating decimal)

Converting Decimals to Fractions

Write the decimal digits over the appropriate power of 10, then simplify.

\(0.6 = \dfrac{6}{10} = \dfrac{3}{5}\) \(0.125 = \dfrac{125}{1000} = \dfrac{1}{8}\)

6. Percents

A percent means "per hundred." The symbol is %.

\[ \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100 \]

Key Conversions

Percent → Decimal: divide by 100 (35% = 0.35)
Decimal → Percent: multiply by 100 (0.07 = 7%)
Fraction → Percent: \(\dfrac{3}{5} = 0.6 = 60\%\)
Percent → Fraction: \(25\% = \dfrac{25}{100} = \dfrac{1}{4}\)

Three Percent Problem Types

Type 1 — Find the Part: What is 30% of 80?
\(0.30 \times 80 = 24\)

Type 2 — Find the Percent: 12 is what percent of 48?
\(\dfrac{12}{48} \times 100 = 25\%\)

Type 3 — Find the Whole: 15 is 60% of what number?
\(\dfrac{15}{0.60} = 25\)

7. Ratios and Proportions

A ratio compares two quantities. A proportion states that two ratios are equal.

Cross-Multiplication
If \(\dfrac{a}{b} = \dfrac{c}{d}\), then \(ad = bc\).

Solve: \(\dfrac{3}{5} = \dfrac{x}{20}\)
\(5x = 60 \Rightarrow x = 12\)

⚠ Common Mistake

Never add denominators: 1/2 + 1/3 ≠ 2/5. Only numerators change when adding fractions — the denominator tells you the piece size, which stays the same.

8. Practice Problems

Simplify \(\dfrac{36}{48}\).
Calculate \(\dfrac{2}{3} + \dfrac{5}{6}\).
Calculate \(\dfrac{7}{8} - \dfrac{1}{3}\).
Calculate \(\dfrac{5}{6} \times \dfrac{9}{10}\).
Calculate \(\dfrac{4}{5} \div \dfrac{8}{15}\).
Convert 0.375 to a fraction in simplest form.
What is 45% of 120?
A class of 32 students has 20 who passed an exam. What percent passed?

Answers

\(\dfrac{3}{4}\)
\(\dfrac{4}{6} + \dfrac{5}{6} = \dfrac{9}{6} = \dfrac{3}{2} = 1\dfrac{1}{2}\)
\(\dfrac{21}{24} - \dfrac{8}{24} = \dfrac{13}{24}\)
\(\dfrac{45}{60} = \dfrac{3}{4}\)
\(\dfrac{4}{5} \times \dfrac{15}{8} = \dfrac{60}{40} = \dfrac{3}{2} = 1\dfrac{1}{2}\)
\(0.375 = \dfrac{375}{1000} = \dfrac{3}{8}\)
\(0.45 \times 120 = 54\)
\(\dfrac{20}{32} \times 100 = 62.5\%\)

Pre-Test Checklist

Addition/Subtraction: find common denominator (LCD) first
Multiplication: multiply numerator × numerator, denominator × denominator
Division: flip the second fraction (reciprocal) then multiply
Always simplify (reduce) your answer

🧠

Spaced Repetition — Ebbinghaus Curve

Review this material at increasing intervals to commit it to long-term memory.

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