Basic Arithmetic Operations

Order of Operations (BODMAS/PEMDAS)

The BODMAS rule tells you the exact order to evaluate any arithmetic expression. Without it, the same expression can give different results depending on who calculates it.

$$8 + 4 \times 3 - 12 \div (6 - 2) = 17$$

Brackets first, then Orders (powers/roots), then Division and Multiplication left-to-right, then Addition and Subtraction left-to-right.

$B$=Brackets — evaluate everything inside first(first priority)

$O$=Orders — powers, exponents, roots(second priority)

$D/M$=Division and Multiplication — equal priority, left to right(third priority)

$A/S$=Addition and Subtraction — equal priority, left to right(fourth priority)

$D$ and $M$ appear together

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Evaluate whichever comes first from left to right — they have equal priority.

$A$ and $S$ appear together

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Same rule — left to right, equal priority.

Left-to-Right Rule: When operations have equal priority (like $\times$ and $\div$, or $+$ and $-$), always work from left to right.

Nested Brackets: If brackets contain other brackets, evaluate the innermost brackets first and work outward.

PEMDAS Reminder: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction — same rule, different acronym.

Evaluating Complex Expressions

When an expression combines multiple operations, apply BODMAS systematically: resolve all brackets first, then powers, then multiply/divide, then add/subtract.

$$3 \times (2 + 4)^2 - 16 \div 4 + 7 = 107$$

Brackets: $(2 + 4) = 6$. Orders: $6^2 = 36$. Multiply: $3 \times 36 = 108$. Divide: $16 \div 4 = 4$. Add/subtract: $108 - 4 + 7 = 111$.

$\text{Step 1}$=Brackets: $(2 + 4) = 6$(—)

$\text{Step 2}$=Orders: $6^2 = 36$(—)

$\text{Step 3}$=Multiply: $3 \times 36 = 108$(—)

$\text{Step 4}$=Divide: $16 \div 4 = 4$(—)

$\text{Step 5}$=Subtract and Add: $108 - 4 + 7 = 111$(—)

Multiple bracket levels

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Start with the innermost bracket and work outward.

Write Each Step: When simplifying, write out each step on a separate line to avoid errors.

Check with Reverse Operations: After calculating, verify by working backward — if $108 - 4 + 7 = 111$, then $111 - 7 + 4 = 108$, which is $3 \times 36 = 3 \times 6^2 = 3 \times (2+4)^2$. ✓

Fraction Fundamentals

A fraction represents a part of a whole. The numerator (top) tells you how many parts you have, and the denominator (bottom) tells you how many equal parts the whole is divided into.

$$\frac{a}{b} \text{ where } b \neq 0$$

Every fraction has a numerator divided by a non-zero denominator.

$a$=Numerator — the number of parts taken(—)

$b$=Denominator — the total number of equal parts(—)

$a = b$

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The fraction equals 1.

$a = 0$

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The fraction equals 0.

$b = 0$

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Undefined — division by zero is never allowed.

Proper Fraction: numerator < denominator, e.g. $\frac{3}{5}$. Value is less than 1.

Improper Fraction: numerator ≥ denominator, e.g. $\frac{7}{4}$. Value is 1 or more.

Mixed Number: A whole number combined with a proper fraction, e.g. $1\frac{3}{4} = \frac{7}{4}$.

[Simplification] Rules: Divide both numerator and denominator by their HCF to get the simplest form.

Adding and Subtracting Fractions

To add or subtract fractions, they must share a common denominator. When the denominators differ, find the LCM and convert both fractions before combining.

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Cross-multiply to create a common denominator, then add the numerators.

$ad$=First numerator × second denominator(—)

$bc$=Second numerator × first denominator(—)

$bd$=Common denominator (product of both)(—)

Same denominator

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Just add numerators: $\frac{a}{d} + \frac{c}{d} = \frac{a+c}{d}$.

Finding the LCD: Instead of multiplying denominators, find their LCM for smaller numbers. E.g., for $\frac{1}{6} + \frac{1}{8}$, use LCD = 24, not 48.

Always Simplify: After adding, reduce the result by dividing numerator and denominator by their HCF.

Mixed Numbers: Convert to improper fractions first, then add/subtract, then convert back.

Multiplying and Dividing Fractions

Multiplying fractions is straightforward — multiply numerators together and denominators together. No common denominator needed.

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

Multiply straight across — numerator × numerator, denominator × denominator.

$a \times c$=Product of numerators(—)

$b \times d$=Product of denominators(—)

Cross-cancellation possible

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Before multiplying, simplify any numerator with any denominator that share a common factor.

Cross-Cancel First: Before multiplying $\frac{4}{9} \times \frac{3}{8}$, cancel 4 with 8 (gives 1 and 2) and 3 with 9 (gives 1 and 3). Result: $\frac{1}{6}$.

Multiplying by a Whole Number: Write the whole number as $\frac{n}{1}$, then multiply. E.g., $5 \times \frac{2}{3} = \frac{10}{3} = 3\frac{1}{3}$.

To divide by a fraction, multiply by its reciprocal. Never divide fractions directly — always flip and multiply.

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

Flip the second fraction (take its reciprocal), then multiply.

$\frac{d}{c}$=Reciprocal of $\frac{c}{d}$(—)

Dividing by a whole number

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Write it as a fraction: $\frac{a}{b} \div n = \frac{a}{b} \times \frac{1}{n}$.

Why Flip?: Division asks 'how many times does the second fit into the first?' Multiplying by the reciprocal gives the same answer.

Keep-Change-Flip: Keep the first fraction, Change $\div$ to $\times$, Flip the second fraction.

Decimal Operations

Decimals are another way to represent parts of a whole. Each position after the decimal point represents a power of ten: tenths, hundredths, thousandths, and so on.

$$0.375 = \frac{3}{10} + \frac{7}{100} + \frac{5}{1000} = \frac{375}{1000} = \frac{3}{8}$$

Read each digit according to its place value and sum them as fractions.

$0.3$=Three tenths = $\frac{3}{10}$(—)

$0.07$=Seven hundredths = $\frac{7}{100}$(—)

$0.005$=Five thousandths = $\frac{5}{1000}$(—)

Terminating decimal

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Has a finite number of digits after the decimal point (e.g., 0.25).

Repeating decimal

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Has digits that repeat forever (e.g., $\frac{1}{3} = 0.\overline{3}$).

Decimal to Fraction: Write the decimal without the point as the numerator, and $10^n$ (where $n$ is the number of decimal places) as the denominator. Then simplify.

Fraction to Decimal: Divide the numerator by the denominator. E.g., $\frac{3}{8} = 3 \div 8 = 0.375$.

Place Value Matters: In 0.452, the 4 is in the tenths place, 5 in hundredths, 2 in thousandths.

Operations with Decimals

Adding and subtracting decimals requires aligning the decimal points. Multiplication counts total decimal places, and division shifts the decimal to make the divisor a whole number.

$$3.14 \times 2.5 = 7.85$$

Multiply as whole numbers (314 × 25 = 7850), then place the decimal point so there are 2 + 1 = 3 decimal places: 7.850.

$\text{2 places}$=Decimal places in 3.14(—)

$\text{1 place}$=Decimal places in 2.5(—)

$\text{3 places}$=Total decimal places in the product(—)

Multiplying by 10, 100, 1000

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Move the decimal point 1, 2, or 3 places to the right.

Dividing by 10, 100, 1000

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Move the decimal point 1, 2, or 3 places to the left.

Multiplication Rule: Ignore the decimal points, multiply as whole numbers, then count total decimal places in both factors and place the point accordingly.

Division Trick: To divide $4.2 \div 0.6$, multiply both by 10 to get $42 \div 6 = 7$.

Addition/Subtraction: Stack vertically with decimal points aligned, then add/subtract column by column.

Percentage Fundamentals

A percentage is a fraction out of 100. The percent symbol (%) means 'per hundred', so 25% = $\frac{25}{100} = \frac{1}{4} = 0.25$.

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

To find what percentage one number is of another, divide the part by the whole and multiply by 100.

$\text{Part}$=The portion or amount you're interested in(same as Whole)

$\text{Whole}$=The total or reference amount(—)

$\times 100$=Converts the decimal to a percentage(—)

Part > Whole

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The percentage exceeds 100% — e.g., 150 out of 100 is 150%.

Whole = 0

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Undefined — you cannot calculate a percentage of zero.

Percentage to Decimal: Divide by 100 (move decimal point 2 places left). 45% = 0.45.

Decimal to Percentage: Multiply by 100 (move decimal point 2 places right). 0.82 = 82%.

Percentage to Fraction: Put over 100 and simplify. 60% = $\frac{60}{100} = \frac{3}{5}$.

Fraction to Percentage: Divide numerator by denominator, multiply by 100. $\frac{3}{8} = 0.375 \times 100 = 37.5%$.

Percentage Calculations

Finding $x$% of a number is the most common percentage operation. The word 'of' translates directly to multiplication.

$$x\% \text{ of } y = \frac{x}{100} \times y$$

Convert the percentage to a decimal (÷100), then multiply by the number.

$x$=The percentage value(%)

$y$=The number you're finding the percentage of(—)

$\frac{x}{100}$=The percentage written as a decimal(—)

$x = 100$

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100% of any number is the number itself.

$x = 50$

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50% is just half — divide by 2.

Quick Mental Tricks: 10% = move decimal one place left. 1% = move two places left. 5% = half of 10%. 25% = divide by 4.

Finding the Whole: If 30% of a number is 45, the number = $45 \div 0.30 = 150$.

Percent Change: $\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$. Positive = increase, negative = decrease.

Mixed Operations and Complex Simplification

Real-world problems often combine fractions, decimals, and percentages in a single calculation. Convert everything to the same form first, then apply BODMAS.

$$\frac{2}{5} \times 0.25 + 30\% \text{ of } \frac{4}{3}$$

Convert all terms to fractions: $\frac{2}{5} \times \frac{1}{4} + \frac{30}{100} \times \frac{4}{3} = \frac{1}{10} + \frac{2}{5} = \frac{1}{10} + \frac{4}{10} = \frac{5}{10} = \frac{1}{2}$.

$\frac{2}{5}$=Fraction (already in fraction form)(—)

$0.25$=Decimal = $\frac{1}{4}$(—)

$30\%$=Percentage = $\frac{3}{10}$(—)

Choosing which form to use

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Fractions are usually easiest for multiplication/division. Decimals for addition/subtraction with money.

Convert to One Form: When mixing types, convert everything to fractions (usually easiest) before operating.

Simplify at Each Step: After each operation, reduce fractions to keep numbers small and manageable.

Estimate First: Before calculating, estimate: $\frac{2}{5} \approx 0.4$, so $0.4 \times 0.25 \approx 0.1$, and $30\% \times 1.3 \approx 0.4$. Total $\approx 0.5$.