ICAP QAT flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Successive percentage change

Two successive changes of a% and b% equal a + b + (ab/100) percent overall. A 20% rise then 20% fall gives 20 − 20 − 400/100 = −4%, a net 4% decrease.

Profit percentage base

Profit or loss percent is calculated on cost price unless the question states otherwise. Selling price equals cost price multiplied by (100 + profit%)/100, or (100 − loss%)/100 for a loss.

Discount base

Discount is always a percentage of the marked price, so selling price = marked price × (100 − d)/100. A 25% discount on a marked price of 800 yields a selling price of 600.

CI minus SI for two years

For two years at rate r, compound interest exceeds simple interest by P(r/100)². On 10,000 at 10% the gap is 10,000 × 0.01 = 100.

Simple interest formula

Simple interest equals principal × rate × time ÷ 100, with rate as a percent per year and time in years. Interest is charged only on the original principal, never on accrued interest.

Ratio share of total

To divide a total T in the ratio a:b:c, first add the parts to get a + b + c, then each share is T times its own part divided by that sum.

Partnership profit split

In a partnership, profit is shared in the ratio of capital multiplied by time invested. A partner investing 5,000 for 12 months and another 10,000 for 6 months share equally.

Weighted average

The weighted average equals the sum of each value multiplied by its weight, divided by the total of the weights. It lies between the smallest and largest values being averaged.

Average shift on replacement

If one value in a set of n changes from x to y, the average changes by (y − x)/n. Replacing 40 with 60 in a group of 5 raises the average by 4.

km/h to m/s

Multiply a speed in kilometres per hour by 5/18 to convert it to metres per second. Thus 72 km/h equals 72 × 5/18 = 20 m/s, a conversion used in train problems.

Relative speed

For two bodies moving in opposite directions, relative speed is the sum of their speeds; in the same direction it is the difference. Time to meet equals distance divided by relative speed.

Boats and streams

Downstream speed equals boat speed plus stream speed, upstream equals boat speed minus stream speed. Boat speed is half the sum of downstream and upstream speeds.

Combined work formula

If A alone takes a days and B alone takes b days, working together they finish in ab/(a + b) days. Two workers taking 6 and 12 days together need 72/18 = 4 days.

Pipes and cisterns

An inlet pipe adds to the fill rate and an outlet subtracts, so net rate is the sum of filling rates minus emptying rates. Time to fill equals capacity divided by net rate.

Simple vs compound interest growth

Simple interest grows linearly with time, while compound interest grows geometrically. Over the same period at the same positive rate, compound interest always exceeds simple interest after the first year.

Data interpretation base check

Before computing any percentage in a data set, confirm the base: total, previous period, or a single category. Percentage change uses the earlier value as denominator, not the later one.

Divisibility by 3 and 9

A number is divisible by 3 if its digit sum is divisible by 3, and by 9 if its digit sum is divisible by 9. The digit sum of 4,731 is 15, so it is divisible by 3 but not 9.

Divisibility by 4 and 8

A number is divisible by 4 when its last two digits form a multiple of 4, and by 8 when its last three digits form a multiple of 8. The rest of the digits do not affect the test.

Divisibility by 11

Alternately add and subtract the digits; the number is divisible by 11 if the result is 0 or a multiple of 11. For 2,728 the difference is 2 − 7 + 2 − 8 = −11, so it is divisible.

HCF and LCM product rule

For two positive integers, HCF multiplied by LCM equals the product of the numbers. This lets you find one value when the other, the product, and one of the two is known.

HCF and LCM of fractions

The LCM of fractions is the LCM of the numerators divided by the HCF of the denominators. The HCF of fractions reverses this: HCF of numerators over LCM of denominators.

Counting factors of a number

Write the number as a product of prime powers p^a·q^b·r^c; the number of factors is (a+1)(b+1)(c+1). For 72 = 2³ × 3² the count is 4 × 3 = 12.

Sum of factors formula

For N = p^a·q^b, the sum of all factors is [(p^(a+1) − 1)/(p − 1)] × [(q^(b+1) − 1)/(q − 1)]. Each prime contributes a geometric series of its powers.

Remainder of a product

To find (a × b) mod d, reduce each factor modulo d first, multiply the remainders, then reduce again. This avoids working with large numbers in remainder questions.

Fermat's little theorem

If p is prime and a is not a multiple of p, then a^(p−1) leaves remainder 1 when divided by p. It simplifies large powers in remainder problems modulo a prime.

Circle area and circumference

Area of a circle is πr² and circumference is 2πr, where r is the radius. Diameter is twice the radius, so area can also be written as πd²/4.

Triangle area formulas

Triangle area equals half the base times the height, or half the product of two sides times the sine of the included angle. Heron's formula uses the semi-perimeter for three known sides.

Sphere and cylinder volume

Sphere volume is (4/3)πr³ with surface area 4πr². A right circular cylinder has volume πr²h and curved surface area 2πrh, excluding the two circular ends.

Scaling of area and volume

If every linear dimension is multiplied by k, areas multiply by k² and volumes by k³. Doubling a cube's edge increases surface area fourfold and volume eightfold.

Permutation versus combination

Permutations count ordered arrangements using nPr = n!/(n−r)!, while combinations count unordered selections using nCr = n!/[r!(n−r)!]. Choosing a team is a combination; assigning seats is a permutation.

Probability addition rule

For two events, P(A or B) equals P(A) + P(B) − P(A and B). Subtract the overlap because outcomes in both events are otherwise counted twice.

Calendar weekday shift

An ordinary year shifts the weekday of a given date by one day, and a leap year shifts it by two. A century year is a leap year only when divisible by 400.