17.1One-page formula sheet#
In brief
Topic Key formulas and facts Reasoning Deductive: general to particular, certain. Inductive: particular to general, probable. Abductive: best explanation. Valid/invalid for…
| Topic | Key formulas and facts |
|---|---|
| Reasoning | Deductive: general to particular, certain. Inductive: particular to general, probable. Abductive: best explanation. Valid/invalid for deduction; strong/weak for induction |
| Series sums | n(n+1)/2; n squared for odd numbers; n(n+1) for even numbers; sum of squares n(n+1)(2n+1)/6; sum of cubes [n(n+1)/2] squared |
| AP and GP | AP: a + (n-1)d; sum = n/2 [2a + (n-1)d]. GP: a r^(n-1); sum = a(r^n - 1)/(r - 1) |
| Alphabet | A=1 ... Z=26; opposite letters sum to 27; EJOTY = 5, 10, 15, 20, 25 |
| Divisibility | 3 and 9 by digit sum; 11 by alternating sums; 4 by last two digits; 8 by last three digits |
| HCF and LCM | HCF x LCM = product (two numbers only); same remainder r: HCF of (a - r), (b - r); smallest number leaving r: LCM + r |
| Factors | N = p^a q^b ..., number of factors = (a+1)(b+1)... |
| Unit digits | Last-digit cycles: 2 gives 2, 4, 8, 6; 3 gives 3, 9, 7, 1; 7 gives 7, 9, 3, 1; 8 gives 8, 4, 2, 6; 4 gives 4, 6; 9 gives 9, 1 |
| Ratio | Share = N x a/(a+b+c); fourth proportional bc/a; mean proportional square root of (ab) |
| Alligation | Cheaper : dearer = (d - m) : (m - c) |
| Percentage | Change % = (new - old)/old x 100; successive: a + b + ab/100; A x% more than B, B is x/(100+x) x 100 % less than A |
| Profit and loss | Profit % = profit/CP x 100; SP = CP(100 + P)/100; discount on MP; successive discount d1 + d2 - d1 d2/100; same-SP gain and loss x%: loss x squared/100 % |
| Simple interest | PRT/100; A = P(1 + RT/100) |
| Compound interest | A = P(1 + R/100)^n; half-yearly: R/2 and 2n; CI - SI (2 years) = P(R/100) squared |
| Discounting | PW = FV x 100/(100 + RT); TD = SI on PW; BD = SI on FV; BG = TD squared/PW |
| Average | Sum = average x n; combined = (n1 a1 + n2 a2)/(n1 + n2); new item = old average + (n+1) x change |
| Time and work | Rate = 1/x; together xy/(x + y); wages in the ratio of work done; pipes: inlet +, outlet - |
| Speed | km/h x 5/18 = m/s; average speed (equal distances) 2ab/(a + b); relative speed: opposite add, same subtract |
| Trains | Pole: L/S; platform: (L + P)/S; two trains: (L1 + L2)/relative speed |
| Boats | Down = u + v; up = u - v; u = (D + U)/2; v = (D - U)/2 |
| Mensuration | Circle pi r squared and 2 pi r; cylinder pi r squared h; cone 1/3 pi r squared h; sphere 4/3 pi r cubed |
| PnC and probability | nPr = n!/(n-r)!; nCr = n!/[r!(n-r)!]; P(E) = favourable/total; P(not E) = 1 - P(E) |