BCS Written Math & Mental Ability cheat sheet

Divisibility: 3 and 9 by digit sum; 4 by last two digits; 8 by last three digits; 11 by alternating digit-sum difference being 0 or a multiple of 11.

HCF x LCM = product of the two numbers, valid only for exactly two positive integers.

HCF of fractions = HCF of numerators / LCM of denominators; LCM of fractions = LCM of numerators / HCF of denominators.

To compare fractions a/b and c/d, cross-multiply: a/b > c/d exactly when ad > bc (b, d positive).

A terminating decimal occurs only when the reduced denominator has no prime factors other than 2 and 5.

Converting repeating decimal 0.abc-bar to a fraction: numerator = abc, denominator = 999 (shift by 10^k for mixed repeats).

Euclid's algorithm: HCF(a, b) = HCF(b, a mod b), repeated until the remainder is 0.

Number of factors of N = p^a q^b ... is (a+1)(b+1)...; sum of factors = product of (p^(a+1)-1)/(p-1).

x% of N = (x/100) x N; to convert a fraction to percent multiply by 100.

Percentage change = (change / original value) x 100; the original is always the base, not the new value.

Successive changes of a% and b% give a net change of a + b + ab/100 percent.

If A is r% more than B, then B is (100r/(100+r))% less than A.

Price increased by r% then decreased by r% gives a net loss of r^2/100 percent.

Population/value after n years at r% = P(1 + r/100)^n; depreciation uses (1 - r/100)^n.

To express a:b as percentages of the total, use 100a/(a+b) and 100b/(a+b).

Profit = SP - CP; Loss = CP - SP; both percentages are always computed on cost price unless stated otherwise.

SP = CP(100 + profit%)/100 and SP = CP(100 - loss%)/100.

CP = 100 x SP/(100 + profit%) or 100 x SP/(100 - loss%).

Equal selling prices with a gain of x% on one item and a loss of x% on another always give a net loss of x^2/100 percent.

Marked price with discount d%: SP = MP(1 - d/100); profit% = (SP - CP)/CP x 100.

If an article is sold at a gain of x% and the SP is reduced by a fixed amount, recompute CP from the new SP before comparing.

False weight claims: gain% = (true weight - false weight)/false weight x 100.

Discount = Marked Price - Selling Price; discount% is always on the marked price.

Single equivalent discount for successive discounts a% and b% = a + b - ab/100 percent.

SP = MP x (100 - d1)/100 x (100 - d2)/100 for two successive discounts.

Profit% after a discount d% when markup is m%: profit% = [(100+m)(100-d)/100] - 100.

To earn a profit of p% after allowing discount d%, MP must be CP x (100+p)/(100-d).

Discount is a reduction on MP; profit or loss is measured against CP, so the two percentages use different bases.

Buy-x-get-y-free means the effective discount is y/(x+y) x 100 percent.

Ratio a:b scales to ka:kb; combine ratios by making the shared term equal (a:b and b:c give a:b:c).

Proportion a:b = c:d means ad = bc; the mean proportional between a and c is the square root of ac.

Divide a quantity Q in the ratio a:b:c using parts Qa/(a+b+c), etc.

Partnership profit shares are proportional to capital x time invested.

If capitals are in ratio a:b and periods in ratio x:y, profit shares are in ratio ax:by.

Compounded ratio of a:b and c:d is ac:bd; duplicate ratio of a:b is a^2:b^2.

In a mixture ratio change, adding only one component keeps the other component's quantity fixed, so equate that term.

Work rate is the reciprocal of time: A alone in x days works at 1/x per day.

Combined rate = sum of individual rates; time together = 1/(1/x + 1/y) = xy/(x+y).

If A is k times as efficient as B, A takes 1/k of B's time and their times are in ratio 1:k.

Work done = rate x time; remaining work after n days = 1 - n(combined rate).

Pipes: an inlet adds its rate, an outlet subtracts its rate; net rate = sum of inlets - sum of outlets.

If a pipe fills in x hours and a leak empties in y hours, the net filling time is xy/(y - x), valid when y > x.

MDH/W constant: (M1 D1 H1)/W1 = (M2 D2 H2)/W2 for men, days, hours and work.

Two workers alternating days: compute the 2-day cycle output, then divide total work by it and handle the remainder.

Speed = Distance/Time; convert km/h to m/s by multiplying by 5/18, and m/s to km/h by 18/5.

Average speed for equal distances at speeds u and v is 2uv/(u+v), not the arithmetic mean.

A train of length L crossing a pole takes L/speed; crossing a platform of length P takes (L+P)/speed.

Two trains of lengths L1, L2 moving opposite directions have relative speed u+v; same direction, u-v.

Time to cross each other = (L1 + L2)/(relative speed), with lengths in metres and speed in m/s.

Upstream speed = boat - stream; downstream = boat + stream; boat speed = (down + up)/2.

Distance is unchanged when start time shifts, so equate d = s1 t1 = s2 t2 to find the time lost or gained.

Average = sum of values / number of values; sum = average x count.

If every value is increased by k, the average increases by k; if multiplied by k, the average multiplies by k.

Combined average of groups = (n1 a1 + n2 a2)/(n1 + n2), a weighted mean.

When one item is replaced, change in average x count = difference between the new and old item.

Average of first n natural numbers = (n+1)/2; of their squares = (n+1)(2n+1)/6.

Average speed over equal distances is the harmonic-type mean 2uv/(u+v), not (u+v)/2.

In an arithmetic sequence the average equals the middle term when the count is odd, or the mean of the two middle terms.

Alligation rule: ratio of quantities = (mean price - cheaper price) : (dearer price - mean price).

Repeated replacement: if a vessel of volume V has pure liquid and x litres are removed and replaced by water n times, remaining pure liquid = V(1 - x/V)^n.

Mixing two solutions of concentration c1 and c2 in ratio r1:r2 gives concentration (r1 c1 + r2 c2)/(r1 + r2).

To dilute a mixture to a target concentration, add water so that the solute quantity stays constant.

Milk:water ratio a:b; adding k litres of water gives a new ratio a:(b+k) only if total volume changes accordingly.

Alligation only applies to two components; with three, pair the cheapest with the dearest first, then mix with the middle.

Cost price of a mixture = total cost of components / total quantity mixed.

SI = PRT/100, with T in years; rate and time must use matching units.

Amount under SI = P(1 + RT/100); under CI = P(1 + R/100)^T.

CI - SI for 2 years = P(R/100)^2; for 3 years the difference is P(R/100)^2 (3 + R/100).

Half-yearly compounding: use rate R/2 and periods 2T; quarterly uses R/4 and 4T.

Effective annual rate for nominal R compounded n times a year = (1 + R/n)^n - 1.

Doubling under SI: T = 100/R years; under CI the rule of 72 estimates doubling time as 72/R years.

If an amount becomes x times in T years under CI, it becomes x^k times in kT years.

For ax + by = c and dx + ey = f, solve by elimination; unique solution when ae - bd is not 0.

Quadratic ax^2 + bx + c = 0 has sum of roots -b/a and product c/a; discriminant b^2 - 4ac decides real roots.

Multiplying or dividing an inequality by a negative number reverses the inequality sign.

|x| < a means -a < x < a; |x| > a means x < -a or x > a.

Arithmetic sequence: nth term a + (n-1)d, sum n/2 [2a + (n-1)d].

Geometric sequence: nth term ar^(n-1), sum a(r^n - 1)/(r - 1) for r not equal to 1; infinite sum a/(1-r) when |r| < 1.

Sum of first n natural numbers = n(n+1)/2; sum of first n odd numbers = n^2.

If x + 1/x = k, then x^2 + 1/x^2 = k^2 - 2 and x^3 + 1/x^3 = k^3 - 3k.

Triangle area = (1/2) x base x height = square root of s(s-a)(s-b)(s-c) with s the semi-perimeter.

Pythagoras: in a right triangle, hypotenuse^2 = sum of squares of the other two sides; common triples are 3-4-5, 5-12-13, 8-15-17.

Circle: area = pi r^2, circumference = 2 pi r; sector area = (theta/360) pi r^2, arc length = (theta/360) 2 pi r.

Rectangle: area = lb, perimeter = 2(l+b); diagonal = square root of (l^2 + b^2).

Cylinder: volume = pi r^2 h, curved surface = 2 pi r h, total surface = 2 pi r(r + h).

Cone: volume = (1/3) pi r^2 h, curved surface = pi r l with slant height l = square root of (r^2 + h^2).

Sphere: volume = (4/3) pi r^3, surface area = 4 pi r^2; hemisphere total surface = 3 pi r^2.

Sum of interior angles of an n-sided polygon = (n - 2) x 180 degrees; each exterior angle of a regular n-gon = 360/n degrees.

Check differences first, then ratios, then alternating or two interleaved series when differences are irregular.

Squares, cubes, primes and their neighbours are common series bases: 1, 4, 9, 16... or 2, 3, 5, 7, 11...

Letter positions A=1 to Z=26; opposite letters sum to 27 (A-Z, B-Y, C-X, and so on).

In coding, compare plain and coded letters position by position to find a shift (+1, -2) or reversal rule.

Number coding often uses digit sums, products, or reversal; test the rule on every digit before accepting it.

Analogy: identify the relation (part-whole, cause-effect, synonym, unit conversion) and apply the same relation to the pair.

Odd one out: test each option against a single property such as primality, divisibility, square status, or letter pattern.

Alphabet series: write positions above letters and convert jumps into numeric differences to reveal the pattern.