ICAP QAT cheat sheet

Percentage change = (new − old) ÷ old × 100; successive changes of a% then b% compound as a + b + ab/100, not a + b.

Profit/loss percentage is always computed on cost price unless stated otherwise; selling price = CP × (100 ± p)/100. Discount is computed on marked price: SP = MP × (100 − d)/100.

Simple interest = PRT/100; compound interest = P[(1 + r/100)^n − 1]. For 2 years the CI − SI gap equals P(r/100)².

Ratio–proportion: if a:b = c:d then ad = bc. To split a total T in ratio a:b:c, shares are T·a/(a+b+c), etc. Partnership profit splits by capital × time.

Average = sum ÷ count. If one item changes from x to y in a set of n, the average shifts by (y − x)/n. Weighted average = Σwᵢxᵢ ÷ Σwᵢ.

Time–speed–distance: distance = speed × time. Convert km/h to m/s by multiplying by 5/18. Relative speed adds for opposite directions, subtracts for same direction.

Boats and streams: downstream speed = b + s, upstream = b − s, so boat speed b = (down + up)/2 and stream s = (down − up)/2.

Work: if A finishes in a days and B in b days, together they take ab/(a+b) days. Use the LCM method to convert rates to units per day.

Data interpretation: read axis scales and units before computing; verify percentage questions against the base stated (total, previous year, or category), since the wrong base is the most common trap.

Divisibility: by 3 or 9 if digit sum is divisible; by 4 if last two digits are; by 8 if last three digits are; by 11 if the alternating digit-sum difference is 0 or a multiple of 11.

HCF × LCM = product of the two numbers (for two positive integers). LCM of fractions = LCM of numerators ÷ HCF of denominators; HCF of fractions is the reverse.

Remainder theorem: for divisor d, (a × b) mod d = [(a mod d)(b mod d)] mod d. Fermat's little theorem: a^(p−1) ≡ 1 mod p when p is prime and a is not a multiple of p.

Number of factors: if N = p^a·q^b·r^c, the count is (a+1)(b+1)(c+1); the sum of factors is [(p^(a+1)−1)/(p−1)] × … over all primes.

Mensuration: circle area = πr², circumference = 2πr; triangle area = ½ × base × height; trapezium = ½(a+b)h; sphere volume = (4/3)πr³, surface = 4πr²; cylinder volume = πr²h.

When all linear dimensions scale by k, areas scale by k² and volumes by k³ - use this instead of recomputing full formulas.

Permutations nPr = n!/(n−r)! for ordered selection; combinations nCr = n!/[r!(n−r)!] for unordered. nC0 = nCn = 1 and nCr = nC(n−r).

Probability = favourable outcomes ÷ total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B); independent events multiply.

Calendar and clock facts: an ordinary year shifts the weekday by 1, a leap year by 2; clock hands coincide 11 times in 12 hours, and the minute hand gains 5.5° per minute on the hour hand.