Percentages
Change (%) = (new − original) / original × 100
The change formula needs a non-zero original value. For a reduction of p%, multiply by 1 − p/100. To recover the original from a discounted amount, divide by that multiplier.
Example: Rs. 960 after 20% off means 960 ÷ 0.80 = Rs. 1,200 originally. See the complete lesson.
Ratios and proportions
A = Tm/(m+n), B = Tn/(m+n)
Use consistent units and the correct total. A ratio of 3:5 has eight parts, not five. Example: 64 counters split 3:5 give 24 and 40 counters.
Averages
Combined mean = (n₁m₁ + n₂m₂) / (n₁ + n₂)
Weight group means by their group sizes. Averaging two means directly works only when the group sizes are equal. Example: three values averaging 10 and two averaging 20 have combined mean (30 + 40)/5 = 14.
Speed and work rates
Combined work rate = 1/a + 1/b
Time for one job = 1 / combined rate
In the work formula, a and b are the times each worker takes alone for the same job at constant rates. Example: 6 hours and 3 hours give a combined rate of 1/2 job per hour, so the job takes 2 hours.
Average speed is total distance divided by total time. It is not generally the arithmetic mean of two speeds. For equal distances at positive speeds u and v, the average is 2uv/(u+v).
Algebra
(a−b)² = a² − 2ab + b²
a²−b² = (a−b)(a+b)
For ax²+bx+c = 0 with a ≠ 0, x = [−b ± √(b²−4ac)]/(2a). Real roots require b²−4ac ≥ 0. Check the question’s allowed domain and reject values that make a denominator zero.
Example: x²−9 = 0 gives (x−3)(x+3) = 0, so x = 3 or −3.
Geometry
| Shape | Formula | Check |
|---|---|---|
| Rectangle | Area = length × width | Area uses square units. |
| Triangle | Area = ½ × base × perpendicular height | A sloping side is not usually the height. |
| Circle | Area = πr²; circumference = 2πr | Radius is half the diameter. |
| Right triangle | a² + b² = c² | c is the hypotenuse; use only for a right angle. |
| Rectangular solid | Volume = length × width × height | Volume uses cubic units. |
Example: a triangle with base 8 cm and perpendicular height 5 cm has area 20 cm².
Probability and counting
P(A then B) = P(A) × P(B given A)
The simple outcome-count formula assumes equally likely outcomes. For draws without replacement, update the remaining counts. Example: two red balls from a bag with three red and two blue have probability (3/5)(2/4) = 3/10.
For n distinct objects, arrangements of r objects are n!/(n−r)!, while selections where order does not matter are n!/[r!(n−r)!], with integers 0 ≤ r ≤ n. Repetition or identical objects changes the counting problem.
Use the sheet actively
Cover the formulas and write the one you need from memory. Explain its conditions, then solve a fresh example. In the app’s Quant Climb, use the flashcards and the Unknown only button to revisit formulas you cannot yet recall confidently.
Examples on this page are original teaching exercises. For exam dates, eligibility, required stream and official instructions, consult the current HEC notice and ETC portal.
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