How to use this formula sheet

A quantitative aptitude formula is useful only when you recognise the situation in which it applies. Revise this sheet with a notebook: write the formula from memory, solve the example, then change one value and solve again. RBI, EPFO and DRDO preparation may overlap in arithmetic and reasoning, but their notified syllabi and time constraints differ. Use this as a common foundation and select practice sets according to the relevant examination notice.

All examples below are original learning examples. Percentages are expressed as numbers such as 12 for 12% unless the formula explicitly uses a decimal. Keep units consistent, distinguish a percentage change from a percentage-point change, and round only at the end. Many apparent calculation mistakes begin with an incorrect denominator, a mismatched time period or an assumption the question never made.

Percentages, fractions and successive changes

A rise from 40% to 50% is 10 percentage points and a 25% relative increase. For successive changes, assign negative signs to decreases: a 20% increase followed by a 10% decrease produces 20 − 10 − 2 = 8% overall growth. Applying both changes to the original value would give the wrong answer because the second change operates on a new base.

Memorise fraction equivalents such as 1/8 = 12.5%, 1/6 ≈ 16.67%, 1/3 ≈ 33.33% and 3/8 = 37.5%. If expenditure remains unchanged while price rises by 25%, consumption becomes 100/125 of its earlier level, a 20% reduction. This inverse relationship appears in price-consumption and productivity questions.

Ratio, proportion and partnership

If A:B = a:b and the total is T, then A = Ta/(a+b) and B = Tb/(a+b). For A:B = 3:5 with a total of 640, the shares are 240 and 400. A ratio is a comparison, so equivalent ratios represent the same relationship even when the absolute quantities differ.

In direct proportion, x/y stays constant; in inverse proportion, xy stays constant. Partnership profit is divided in the ratio of capital multiplied by time when the agreement bases sharing on those contributions. A invests ₹40,000 for twelve months and B invests ₹60,000 for eight months: their weighted investments are equal, so the profit shares are equal. Account for capital changes in separate time intervals instead of averaging the investments casually.

For combined ratios, first equalise the common term. If A:B = 2:3 and B:C = 4:5, multiply the first ratio by four and the second by three to obtain A:B:C = 8:12:15. This method is safer than multiplying unrelated terms without identifying which quantity they represent.

Averages, weighted means and mixtures

Average = total quantity / number of observations. Combined average = (n₁a₁ + n₂a₂)/(n₁+n₂). If ten students average 50 and twenty average 65, the combined average is 60, not 57.5. The larger group deserves the larger weight.

For two ingredients with values L and H, mixed to obtain M between them, the ratio of low-value quantity to high-value quantity is (H−M):(M−L). Mixing solutions priced at ₹30 and ₹50 per litre to obtain ₹38 per litre gives 12:8, or 3:2. This alligation shortcut assumes quantities are additive and the stated value behaves as a weighted average.

If a fraction x/V of a well-mixed liquid is removed and replaced with another liquid repeatedly, the original liquid remaining after n operations is V(1−x/V)^n. The vessel must be mixed before each removal. Without that condition, the repeated-replacement formula may not describe the process.

Profit, loss, discount and marked price

Profit is calculated on cost price unless the question specifies another base. Discount is calculated on marked price. An item costing ₹800 is marked at ₹1,200 and sold with a 20% discount: selling price is ₹960 and profit is ₹160, or 20%. The discount rate and profit rate happen to be equal here, but they describe different comparisons.

When equal quantities are sold at the same selling price with a gain of x% on one and a loss of x% on the other, the overall loss is x²/100 percent. Check the equal-selling-price condition before using the shortcut. Equal cost prices create a different situation.

Simple interest and compound interest

Simple interest = PRT/100. Amount under annual compound interest = P(1+R/100)^T. For half-yearly compounding with an annual nominal rate R, use R/2 per half-year and 2T periods. Do not halve the time and rate simultaneously in the annual formula.

For ₹10,000 at 10% annually for two years, simple interest is ₹2,000 and compound interest is ₹2,100. The difference is ₹100, which equals P(R/100)² for two years at a constant annual rate. For varying annual rates, multiply the separate annual growth factors.

When a question mentions instalments, identify when each payment occurs and how much time remains for interest to accrue. A payment made at the beginning of a year is not equivalent to one made at the end. Draw a short timeline before calculating; it often saves more time than searching for a memorised instalment shortcut.

Time, work, efficiency and pipes

If A completes a job in a days, A’s daily rate is 1/a. Combined rate for A and B is 1/a + 1/b, so their completion time is ab/(a+b). If A takes twelve days and B eighteen days, they together take 36/5 = 7.2 days. The formula assumes constant independent rates and no interference.

Efficiency is inversely proportional to time for the same work. If A is twice as efficient as B, A needs half as much time. For worker-day questions, compare total effective work using workers × days × hours × efficiency, while keeping the work requirement explicit.

A filling pipe contributes a positive rate and a leak a negative rate. A tank filled in six hours and emptied by a leak in nine hours has a net rate of 1/6−1/9 = 1/18, so filling takes eighteen hours. Never add filling and emptying times directly.

Speed, distance, trains and boats

Speed = distance/time. Convert kilometres per hour to metres per second by multiplying by 5/18; reverse the conversion with 18/5. Relative speeds add for opposite directions and subtract for the same direction. A train crossing a pole covers its own length; crossing a platform covers train length plus platform length.

For equal distances travelled at speeds u and v, average speed is 2uv/(u+v). For equal times, it is (u+v)/2. Travelling equal distances at 40 and 60 km/h gives an average of 48 km/h. The arithmetic mean of 50 km/h would incorrectly give equal weight to unequal travel times.

In still water, boat speed = (downstream speed + upstream speed)/2 and stream speed = (downstream speed − upstream speed)/2. These relationships assume the stream speed and the boat’s still-water speed remain constant during the journeys.

Number systems and algebra

For two positive integers, HCF × LCM = their product. Do not generalise this identity to three integers without checking the actual factors. A number divisible by nine has a digit sum divisible by nine; divisibility by eleven depends on the difference between alternating digit sums being divisible by eleven.

Core identities are (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², and a²−b² = (a−b)(a+b). For a quadratic ax²+bx+c = 0, roots sum to −b/a and multiply to c/a. The discriminant b²−4ac identifies whether real roots are distinct, equal or absent.

Use substitution to check an algebraic answer when the options are numerical. Also inspect restrictions: division by x assumes x is non-zero, and squaring both sides can introduce extraneous solutions. A quick check in the original equation is part of solving, not an optional extra.

Mensuration and geometry essentials

Use radius rather than diameter in circle formulas, and distinguish vertical height from slant height in a cone. If every linear dimension doubles, area becomes four times and volume eight times. This scaling rule can solve comparison questions without calculating every measurement.

For a right triangle, hypotenuse squared equals the sum of the squares of the other two sides. Recognise triples such as 3–4–5 and 5–12–13, including their multiples. A diagram should display the given relationships; an attractive-looking sketch is not evidence that two lines are equal or perpendicular.

Statistics, probability and data interpretation

Mean uses all observations; median is the central value after ordering; mode is the most frequent value. For even n, median is the average of the two central observations. Range = maximum − minimum. A weighted mean is often the correct method for combining groups shown in a table or chart.

For equally likely outcomes, probability = favourable outcomes/total outcomes. Complement probability is 1−P(A). For independent events, P(A and B) = P(A)P(B); independence must be justified. For two draws without replacement, the second denominator and often the numerator change.

In data interpretation, read units, scale, dates and legends before calculating. A chart in thousands should not be compared directly with a table in lakh. For percentage growth, use the earlier period as the denominator. To compare ratios quickly, cross-multiply positive quantities or use sensible approximations, but retain enough precision when answer choices are close.

A seven-day formula revision drill

Spend the first two days on percentages, ratio, averages and interest; the next two on work, speed and profit; the fifth on algebra and mensuration; and the sixth on statistics and interpretation. On day seven, solve a mixed timed set and list every error as formula selection, unit conversion, arithmetic or reading.

Retest the weakest category with new numbers rather than repeatedly solving the same remembered example. Keep one page for conditions and exceptions, such as equal distance versus equal time. A formula sheet becomes a scoring resource when it helps you choose a correct method quickly and reject an invalid shortcut confidently.