Rational Numbers and Arithmetic Operations

• Example: For $\frac{3}{8}$, calculate $3 \div 8 = 0.375$.

To convert that decimal to a percentage, multiply it by $100$.

• Example: $0.375 \times 100 = 37.5\%$.

2. Decimals to Fractions and Percentages

To convert a decimal to a percentage, multiply by $100$ (or move the decimal point two places to the right). Conversely, to convert a percentage to a decimal, divide by $100$ (move the decimal two places to the left).

To convert a decimal to a fraction, place the number over its corresponding place value and simplify.

• Example: $0.48$ has its last digit in the hundredths place, so it becomes $\frac{48}{100}$.

3. Percentages to Fractions and Decimals

To convert a percentage to a decimal, divide by $100$. To convert to a fraction, place the percentage value over $100$ and simplify.

• Example: $0.6\% = \frac{0.6}{100}$. To remove the decimal, multiply both top and bottom by $10$ to get $\frac{6}{1000}$, which simplifies to $\frac{3}{500}$.

Exam Tip: When comparing numbers in different formats, it is often most efficient to convert everything into decimal form first.

Comparing and Ordering Rational Numbers

Once numbers are in a consistent format, you can use inequality symbols to compare them:

• $>$ (Greater than)
• $<$ (Less than)
• $\geq$ (Greater than or equal to)
• $\leq$ (Less than or equal to)

Using Decimals for Comparison

When comparing decimals, line them up vertically by their decimal points and analyze the place value from left to right (starting with the highest value, such as the ones place, then tenths, hundredths, etc.).

Using Common Denominators for Fractions

To compare or order fractions with different denominators, you must create equivalent fractions.

1. Find the least common multiple (LCM) of the denominators.
2. The LCM becomes your least common denominator (LCD).
3. Rewrite each fraction using this LCD.

Worked Example: Compare $\frac{2}{5}$, $\frac{1}{6}$, and $\frac{4}{15}$.

• Multiples of $5$: $5, 10, 15, 20, 25, 30...$
• Multiples of $6$: $6, 12, 18, 24, 30...$
• Multiples of $15$: $15, 30, 45...$
• The LCM is $30$.
• Equivalent fractions: $\frac{2}{5} = \frac{12}{30}$; $\frac{1}{6} = \frac{5}{30}$; $\frac{4}{15} = \frac{8}{30}$.
• Increasing Order: $\frac{1}{6}, \frac{4}{15}, \frac{2}{5}$ (or $\frac{5}{30}, \frac{8}{30}, \frac{12}{30}$).

Arithmetic Operations and PEMDAS

To solve complex mathematical expressions, you must follow the order of operations, commonly known by the mnemonic PEMDAS:

1. Parentheses: Perform all operations inside grouping symbols first.
2. Exponents and Roots: Calculate powers and square roots.
3. Multiplication and Division: Perform these from left to right as they occur.
4. Addition and Subtraction: Perform these from left to right as they occur.

Special Rules for Fractions and Mixed Numbers

• Fraction Bars: A fraction bar acts as a grouping symbol. You must complete all operations in the numerator and all operations in the denominator before performing the final division.
• Mixed Numbers: Before multiplying or dividing, convert mixed numbers (an integer combined with a fraction) into improper fractions.
• Division of Fractions: To divide, multiply the first fraction by the reciprocal (the flipped version) of the second fraction.

Worked Example (Division and Reciprocals): Solve $2 \frac{1}{4} \div \frac{3}{5} + 1 \frac{1}{2}$.

1. Convert to improper fractions: 2 \frac{1}{4} &= \frac{9}{4} \\ 1 \frac{1}{2} &= \frac{3}{2} \end{aligned}$$
2. Perform division by multiplying by the reciprocal: \frac{9}{4} \div \frac{3}{5} &= \frac{9}{4} \times \frac{5}{3} = \frac{45}{12} = \frac{15}{4} \end{aligned}$$ $$&#36;
3. Add using a common denominator: \frac{15}{4} + \frac{3}{2} &= \frac{15}{4} + \frac{6}{4} = \frac{21}{4} \text{ or } 5 \frac{1}{4} \end{aligned}$$ &#36;&#36;&#36; $$&#36;

Note: Calculators do not always follow the order of operations in the way you might expect. It is vital to be competent in performing these calculations by hand to ensure accuracy.