Arithmetic

• A percentage is a fraction out of 100. x% of N = N × x/100.
• Profit and loss percentage are always on the cost price, never the selling price. This single rule accounts for most lost marks in the topic.
• Percentage increase = (increase ÷ original) × 100. Using the new value as the base gives a different, wrong answer.
• To reverse a percentage: if a number grew by 20% to reach 60, then 120% = 60, so 100% = 50. Taking 20% off 60 gives 48 — a different question.

• Adding or subtracting needs a common denominator. Never add the denominators.
• Multiplying needs nothing — multiply across the top and across the bottom.
• Dividing: invert the second fraction and multiply.
• 'of' means multiply. Half of three-fifths is ½ × 3/5 = 3/10.
• To compare fractions, turn them into decimals. A bigger denominator does not mean a bigger fraction.

• In a ratio a : b, the total is a + b parts. Find one part first, then multiply.
• Both quantities must be in the same unit before you simplify. 45 minutes to 2 hours is 45 : 120 = 3 : 8, not 45 : 2.
• Direct proportion — more of one means proportionally more of the other (5 pens ₹75, so 8 pens ₹120).
• Inverse proportion — more of one means less of the other (twice the workers, half the time).

• Write both numbers as products of primes.
• HCF — take the lowest power of each shared prime. LCM — take the highest power of every prime that appears.
• Useful check: HCF × LCM = the product of the two numbers.
• 12 = 2²×3 and 18 = 2×3². HCF = 2×3 = 6. LCM = 2²×3² = 36. And 6 × 36 = 216 = 12 × 18.

• SI = P × R × T ÷ 100. The answer is the interest alone; the amount is P + SI.
• BODMAS — brackets, orders (powers), division and multiplication left to right, then addition and subtraction left to right.
• 144 ÷ 12 × 3 is 36, not 4. Division and multiplication rank equally and go left to right.
• Multiplying decimals: multiply as whole numbers, then count the total decimal places. 2.5 × 0.4 → 25 × 4 = 100, two places, so 1.00.

These are the mistakes, not the rules — each one is something a student who knows the topic still does under time pressure.

1. Taking profit or loss percentage on the selling price. It is always on the cost price.
2. Adding numerators and denominators separately: 2/3 + 3/4 is not 5/7.
3. Simplifying a ratio before putting both parts in the same unit.
4. Doing multiplication before division in BODMAS. They rank equally, left to right.
5. Rounding up when the answer must be a whole item. ₹63 buys twelve ₹5 pencils, not thirteen.