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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
→Evaluate whichever comes first from left to right — they have equal priority.
$A$ and $S$ appear together
→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
→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(—)
$a = b$
→The fraction equals 1.
$a = 0$
→The fraction equals 0.
$b = 0$
→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
→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
→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
→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
→Has a finite number of digits after the decimal point (e.g., 0.25).
Repeating decimal
→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
→Move the decimal point 1, 2, or 3 places to the right.
Dividing by 10, 100, 1000
→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
→The percentage exceeds 100% — e.g., 150 out of 100 is 150%.
Whole = 0
→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(—)
$x = 100$
→100% of any number is the number itself.
$x = 50$
→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
→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 $\approx 0.5$.
Application Strategy
Worked Examples
Problem Statement
Evaluate: $12 + 6 \times (8 - 3)^2 \div 5 - 7$
BODMASorder of operations
Strategic Toolkit
$$\text{BODMAS: } B \to O \to D/M \to A/S$$
Step 1
Brackets
Evaluate the expression inside the brackets first.
$(8 - 3) = 5$
Step 2
Orders
Apply the exponent to the bracket result.
$5^2 = 25$
Step 3
Multiply and Divide (left to right)
First multiply $6 \times 25 = 150$, then divide by 5.
$6 \times 25 = 150$, then $150 \div 5 = 30$
Step 4
Add and Subtract (left to right)
Complete the remaining addition and subtraction.
$12 + 30 - 7 = 35$
Problem Statement
Amina spent $\frac{1}{3}$ of her salary on rent and $\frac{1}{4}$ on groceries. What fraction of her salary remains?
fractionsadditionsubtraction
Strategic Toolkit
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$
Step 1
Find total spent
Add the two fractions. LCD of 3 and 4 is 12.
$\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$
Step 2
Find remaining fraction
Subtract the spent portion from the whole (1).
$1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$
Step 3
Result
Amina has $\frac{5}{12}$ of her salary remaining.
$\frac{5}{12}$ of salary remains
Problem Statement
Convert $\frac{7}{8}$ to a decimal and $0.625$ to a fraction in lowest terms.
decimalsfractionsconversion
Strategic Toolkit
$$\frac{a}{b} = a \div b$$
Step 1
Fraction to Decimal
Divide 7 by 8 using long division.
$7 \div 8 = 0.875$
Step 2
Decimal to Fraction
Write 0.625 as $\frac{625}{1000}$ and simplify by dividing by HCF.
$\frac{625}{1000} = \frac{625 \div 125}{1000 \div 125} = \frac{5}{8}$
Problem Statement
A laptop costs Rs. 48,000. During a sale, it is discounted by 15%. What is the sale price?
percentagediscount
Strategic Toolkit
$$x\% \text{ of } y = \frac{x}{100} \times y$$
Step 1
Find the discount amount
Calculate 15% of 48,000.
$\frac{15}{100} \times 48000 = 0.15 \times 48000 = 7200$
Step 2
Subtract from original
Subtract the discount from the original price.
$48000 - 7200 = 40800$
Step 3
Alternative Method
Pay 85% of original: $0.85 \times 48000 = 40800$.
$\text{Sale price} = \text{Rs. } 40,800$
Problem Statement
Simplify: $\frac{3}{4} \times \frac{2}{5} \div \frac{1}{10} + 0.5$
mixed operationsfractionsdecimals
Strategic Toolkit
$$\text{BODMAS + Convert to one form}$$
Step 1
Convert 0.5 to fraction
Convert the decimal to a fraction for uniform operations.
$0.5 = \frac{1}{2}$
Step 2
Multiply fractions
Multiply $\frac{3}{4} \times \frac{2}{5}$.
$\frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10}$
Step 3
Divide by $\frac{1}{10}$
Flip and multiply: $\frac{3}{10} \div \frac{1}{10} = \frac{3}{10} \times \frac{10}{1}$.
$\frac{3}{10} \times \frac{10}{1} = 3$
Step 4
Add the final term
Add $\frac{1}{2}$ to the result.
$3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2} = 3.5$
Memorize formulas
BODMAS Order
Use this whenever evaluating an expression with multiple operations
Formula
$$B \to O \to D/M \to A/S$ (left to right within same priority)$
Fraction Addition/Subtraction
When fractions have different denominators
Formula
$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
Fraction Multiplication
Multiply straight across — no common denominator needed
Formula
$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Percentage of a Number
'Of' means multiply — convert percentage to decimal first
Formula
$x\% \text{ of } y = \frac{x}{100} \times y$
Percent Change
For increases and decreases relative to the original value
Formula
$\%\text{ change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$
Review
1
Evaluate: $14 - 3 \times 4 + 8 \div 2$
Ans
6
Review
Tap to reveal
Multiply and Divide first
Apply BODMAS: perform multiplication and division before addition and subtraction.
$3 \times 4 = 12$, $8 \div 2 = 4$
Add and Subtract left to right
Now perform the remaining addition and subtraction from left to right.
$14 - 12 + 4 = 6$
2
A bakery uses $\frac{2}{3}$ of a bag of flour for bread and $\frac{1}{4}$ of the same bag for cookies. What fraction of the bag remains?
Ans
$\frac{1}{12}$
Review
Tap to reveal
Add the fractions used
Find the LCD of 3 and 4 (= 12), convert, and add.
$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$
Subtract from the whole
Subtract the total used from 1 whole bag.
$1 - \frac{11}{12} = \frac{1}{12}$
3
A student scored 72 out of 90 on a test. What is the percentage score?
Ans
80.0%
Review
Tap to reveal
Write as a fraction
Express the score as a fraction of the total marks.
$\frac{72}{90}$
Convert to percentage
Divide and multiply by 100.
$\frac{72}{90} \times 100 = 80\%$
4
Evaluate: $(5 + 3)^2 - 4 \times 6 \div 2 + 1$
Ans
53
Review
Tap to reveal
Brackets and Orders
Evaluate the bracket first, then square.
$(5+3) = 8$, $8^2 = 64$
Multiply and Divide
Perform multiplication and division left to right.
$4 \times 6 = 24$, $24 \div 2 = 12$
Add and Subtract
Complete the remaining operations.
$64 - 12 + 1 = 53$
5
A shopkeeper bought 240 eggs. $\frac{1}{8}$ of them broke during transport. He sold $\frac{2}{3}$ of the remaining eggs. How many eggs are left unsold?
Ans
70
Review
Tap to reveal
Find broken eggs
Calculate $\frac{1}{8}$ of 240.
$\frac{1}{8} \times 240 = 30$ broken
Find remaining and sold
Remaining = $240 - 30 = 210$. Sold = $\frac{2}{3} \times 210 = 140$.
$210 - 140 = 70$ unsold
6
What is $0.375 \times 0.4$?
Ans
0.15
Review
Tap to reveal
Count decimal places
0.375 has 3 decimal places, 0.4 has 1.
Total: 4 decimal places
Multiply and place decimal
$375 \times 4 = 1500$. Place decimal point 4 places from the right.
$0.1500 = 0.15$
7
A phone originally costs Rs. 25,000. It is first discounted by 20%, then an additional 10% is taken off the reduced price. What is the final price?
Ans
Rs. 18,000
Review
Tap to reveal
Apply first discount
20% off means paying 80%.
$25000 \times 0.80 = 20000$
Apply second discount
10% off the reduced price means paying 90% of it.
$20000 \times 0.90 = 18000$
8
Simplify: $\frac{5}{6} \div \frac{10}{9} \times \frac{4}{3}$
Ans
$1$
Review
Tap to reveal
Convert division to multiplication
Flip $\frac{10}{9}$ to its reciprocal.
$\frac{5}{6} \times \frac{9}{10} \times \frac{4}{3}$
Cross-cancel and multiply
Cancel common factors, then multiply what remains.
$\frac{1}{2} \times \frac{3}{2} \times \frac{4}{3} = 1$
9
A factory's production increased from 4,500 units in January to 5,400 units in February. What is the percentage increase?
Ans
20%
Review
Tap to reveal
Find the increase
Subtract the original from the new value.
$5400 - 4500 = 900$
Calculate percentage increase
Divide by the ORIGINAL value and multiply by 100.
$\frac{900}{4500} \times 100 = 20\%$
10
If $\frac{3}{5}$ of a number is 42, what is $\frac{2}{7}$ of the same number?
Ans
20
Review
Tap to reveal
Find the original number
Divide 42 by $\frac{3}{5}$ (multiply by reciprocal).
$n = 42 \times \frac{5}{3} = 70$
Calculate $\frac{2}{7}$ of it
Multiply 70 by $\frac{2}{7}$.
$\frac{2}{7} \times 70 = 20$
11
A merchant mixes 30 kg of rice at Rs. 80/kg with 20 kg at Rs. 120/kg. What is the average price per kg of the mixture?
Ans
Rs. 96/kg
Review
Tap to reveal
Calculate total cost
Multiply each type's quantity by its price and sum.
$30 \times 80 + 20 \times 120 = 4800$
Divide by total weight
Average price = total cost / total quantity.
$\frac{4800}{50} = 96$ Rs./kg
12
Simplify: $\left(\frac{1}{2} + \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right)$
Ans
$\frac{5}{8}$
Review
Tap to reveal
Evaluate first bracket
Find the LCD of 2 and 3, then add.
$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
Evaluate second bracket
Subtract $\frac{1}{4}$ from 1.
$1 - \frac{1}{4} = \frac{3}{4}$
Multiply and simplify
Multiply the two results and reduce.
$\frac{5}{6} \times \frac{3}{4} = \frac{15}{24} = \frac{5}{8}$
Question 1 of 12
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1
Evaluate: $14 - 3 \times 4 + 8 \div 2$