Math Tricks

How to Solve Time and Work Problems

Time and work problems are solved by converting each worker’s output into a rate (work done per unit time), adding rates when workers combine, and using total…

How to Solve Percentage Problems

Percentage problems are solved by treating "percent" as "per hundred": x% of a number is (x/100) × number, and a percentage change is (change ÷ original) × 100.

How to Solve Profit and Loss Problems

Profit and loss are always calculated on the cost price: Profit = SP − CP, Loss = CP − SP, and the percentage is (Profit or Loss ÷ CP) × 100.

How to Solve Ratio and Proportion Problems

A ratio compares two quantities (a : b), and a proportion states that two ratios are equal (a : b = c : d); most problems are solved by introducing a common mu…

How to Solve Time, Speed and Distance Problems

Time, speed and distance problems all reduce to one relation: Distance = Speed × Time, rearranged as Speed = Distance ÷ Time or Time = Distance ÷ Speed, with c…

How to Solve Simple and Compound Interest Problems

Simple interest grows linearly on the original principal (SI = P×R×T/100), while compound interest grows on principal plus accumulated interest, giving CI = P×…

How to Solve Averages Problems

The average of a set is the sum of all values divided by the count, and most averages problems are solved by tracking the total sum rather than the average its…

How to Solve Problems on Ages

Problems on ages are solved by assigning variables to present ages, translating "years ago" and "years hence" into simple linear expressions, and forming equat…

How to Solve Permutation and Combination Problems

Permutations count arrangements where order matters (nPr = n!/(n−r)!), while combinations count selections where order does not matter (nCr = n!/(r!(n−r)!)); t…

How to Solve Probability Problems

Probability of an event is favorable outcomes divided by total possible outcomes, P(E) = n(E)/n(S), and most problems are solved by carefully counting both usi…

How to Solve Boats and Streams Problems

Boats and streams problems are solved by adding the stream’s speed to the boat’s speed for downstream travel and subtracting it for upstream travel, then apply…

How to Solve Pipes and Cisterns Problems

Pipes and cisterns problems extend time-and-work: an inlet pipe fills at a positive rate (1/a of the tank per hour), an outlet pipe drains at a negative rate (…

How to Solve Partnership Problems

Partnership problems split profit in the ratio of each partner’s capital multiplied by the time it was invested, so the profit-sharing ratio is capital₁ × time…

How to Solve Mixture and Alligation Problems

Mixture and alligation problems find the ratio in which two ingredients of different values (price, concentration, etc.) must be mixed to hit a target average,…

How to Solve HCF and LCM Problems

HCF (highest common factor) is the largest number dividing all given numbers, LCM (least common multiple) is the smallest number divisible by all of them, and…

How to Solve Calendar Problems

Calendar problems are solved by counting "odd days" — the remainder when total days between two dates are divided by 7 — since the day of the week repeats ever…

How to Solve Clock Problems

Clock problems are solved by treating the minute hand and hour hand as two objects moving at fixed angular speeds — 6° per minute and 0.5° per minute respectiv…

How to Solve Relative Speed Problems

Relative speed is the speed at which the gap between two moving bodies changes, calculated as the sum of their speeds when moving toward each other or in oppos…

How to Solve Circular Track Race Problems

On a circular track of length L, two runners moving at speeds a and b first meet at L/(a−b) if moving in the same direction, or L/(a+b) if moving in opposite d…

How to Solve Average Speed for Two-Part Journeys

Average speed for a journey is always total distance divided by total time, never the simple average of the two speeds; when equal distances are covered at spe…

How to Solve Escalator Speed Problems

Escalator problems are relative-speed problems where the escalator itself contributes a constant speed e (in steps or distance per unit time) that adds to a pe…

How to Solve Man vs Train Crossing Problems

When a train crosses a person walking or running, the train covers its own length L relative to that person at their relative speed, so time = L / (relative sp…

How to Solve Successive Percentage Change Problems

Successive percentage changes never add or subtract directly; two changes of a% and b% combine to a net change of a + b + (ab/100) percent, where a negative si…

How to Solve Discount and Marked Price Problems

Selling price equals marked price minus discount, SP = MP x (1 - discount%/100), and successive discounts must be chained as multiplying factors rather than ad…