1. Percentage Calculation Formulas & Fraction-to-Percentage Equivalents

Percentage increase/decrease: % Change = [(New Value - Old Value) / Old Value] × 100. Successive percentage changes (a% and b%): Net % Change = a + b + (ab / 100). Fraction shortcuts to memorize: 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16.67%, 1/7 = 14.28%, 1/8 = 12.5%, 1/9 = 11.11%, 1/11 = 9.09%, 1/12 = 8.33%. If price rises by r%, consumption must decrease by [r / (100 + r)] × 100 to keep expenditure constant.

2. Profit, Loss & Marked Price (Discount) Formulas

Cost Price (CP), Selling Price (SP), Marked Price (MP): Profit = SP - CP; Loss = CP - SP; Profit % = (Profit / CP) × 100; Loss % = (Loss / CP) × 100. Calculating SP directly: SP = CP × [(100 ± Profit%/Loss%) / 100]. Calculating CP: CP = SP × [100 / (100 ± Profit%/Loss%)]. Discount = MP - SP; Discount % = (Discount / MP) × 100. Two successive discounts d1% and d2%: Single Equivalent Discount = d1 + d2 - (d1 × d2 / 100). Relation between CP and MP: CP / MP = (100 - Discount%) / (100 + Profit%).

3. Simple Interest (SI) vs Compound Interest (CI) & The Difference Trick

Simple Interest for Principal P, rate R% per annum, time T years: SI = (P × R × T) / 100; Total Amount A = P + SI. Compound Interest compounded annually: Amount A = P(1 + R/100)^T; CI = A - P. Compounded half-yearly: replace R with R/2 and T with 2T. The Golden 2-Year Difference Trick: The difference between CI and SI for 2 years is: CI - SI = P(R / 100)². For 3 years: Difference = P(R / 100)² × [(300 + R) / 100].

4. Ratio, Proportion & Alligation Mixture Formula

Direct proportion: a/b = c/d -> a·d = b·c (product of extremes = product of means). Mean proportional of a and b = √(ab). Third proportional to a and b = b²/a. Fourth proportional to a, b, c = (b × c) / a. Rule of Alligation (Mixture of cheaper price c and dearer price d to produce mean price m): (Quantity of Cheaper) / (Quantity of Dearer) = (d - m) / (m - c).

5. Time, Speed & Distance: Relative Speed & Train Formulas

Speed = Distance / Time. Unit conversion: 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h. Average Speed: When distances are equal: Avg Speed = (2xy) / (x + y), where x and y are speeds. Relative Speed: (1) Moving in opposite directions: Relative Speed = S1 + S2; (2) Moving in same direction: Relative Speed = |S1 - S2|. Trains: Time to cross a pole/man = (Length of Train) / Speed. Time to cross a platform of length L_p = (L_train + L_p) / Speed. Boats and Streams: Downstream speed D = u + v; Upstream speed U = u - v (where u is boat speed in still water, v is stream speed). Boat speed u = (D + U)/2; Stream speed v = (D - U)/2.

6. Time & Work Formulas and the Total Work LCM Method

If A completes work in x days and B in y days: A's 1-day work = 1/x; B's 1-day work = 1/y. Combined 1-day work = 1/x + 1/y = (x + y)/xy. Total time taken together = (x × y) / (x + y) days. Total Work LCM Trick: Take LCM of given day values as "Total Units of Work". Divide total units by individual days to get individual efficiencies (units per day). Sum efficiencies to find days = (Total Units) / (Combined Efficiency). Chain Rule: (M1 × D1 × H1) / W1 = (M2 × D2 × H2) / W2.