Mathematics and Reasoning: Arithmetic Shortcuts and Logical Method
Arithmetic that recurs
High-yield chapters are percentage, ratio and proportion, average, profit-loss-discount, simple and compound interest, time-work and pipes, time-speed-distance, mensuration and simplification. Nearly every question is a variation on one base value, so fixing the base as 100 or as the cost price makes the arithmetic transparent.
Reasoning that recurs
Number and letter series, coding-decoding, blood relations, direction sense, order and ranking, syllogism, linear and circular seating, mirror and water images, and calendar-clock basics. In reasoning, draw the arrangement or the family tree rather than holding it in memory; one written diagram converts a guessing question into a checking question.
Worked example (successive change)
A trader marks an article 40% above cost price and allows a 25% discount. Find the profit per cent. Step 1: let CP = 100. Step 2: MP = 100 + 40 = 140. Step 3: discount is always on the marked price, so SP = 140 × (75/100) = 105. Step 4: profit = 105 − 100 = 5 on a base of 100, i.e. 5%. A single net multiplier (1.40 × 0.75 = 1.05) gives the same result faster and is safer under time pressure.
Shortcuts worth memorising
- Difference between CI and SI for 2 years = P(r/100)²; for 3 years = P(r/100)²(3 + r/100).
- Average speed for equal distances at x and y = 2xy/(x + y), never (x + y)/2.
- Clock angle at H hours M minutes = |30H − 5.5M|, then reduced to the smaller angle.
Traps that cost marks
- Applying discount on cost price instead of marked price, or taking profit percentage on selling price.
- Changing the base mid-calculation in percentage questions (successive increase and decrease never cancel).
- In syllogism, treating a possibility conclusion as definitely false; only a definite contradiction makes it invalid.
- In direction sense, turning left or right after facing south and forgetting to rotate the reference axis.