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tags: - cocubes - aptitude - formula-sheet

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Aptitude Formula Sheet

Percentages

Situation	Fast rule
x percent of y	x/100 multiplied by y
Percentage change	(new − old) / old multiplied by 100
Increase a then decrease b	net percent = a − b − ab/100
Successive increases a and b	net percent = a + b + ab/100
If A is x percent more than B	A = B(1 + x/100)
If A is x percent less than B	A = B(1 − x/100)

Important: +20% then −20% gives a net −4%, not zero.

Useful fraction-to-percent conversions:

Fraction	Percent
1/2	50%
1/3	33.33%
1/4	25%
1/5	20%
1/6	16.67%
1/8	12.5%
1/10	10%
1/12	8.33%
1/16	6.25%

Ratio, proportion and averages

• If A:B = a:b and total is T, then A = T multiplied by a/(a+b).
• To combine A:B = a:b and B:C = c:d, make the two B values equal first.
• Average = total/count.
• New average after adding x: new total = old average multiplied by old count + x.
• Weighted average = sum of value multiplied by weight / sum of weights.

Profit, loss and discount

• Profit = SP − CP.
• Profit percent = profit/CP multiplied by 100.
• Loss percent = loss/CP multiplied by 100.
• Discount = MP − SP.
• Discount percent = discount/MP multiplied by 100.
• Two discounts a and b: equivalent discount = a + b − ab/100.

Simple and compound interest

• SI = PRT/100.
• Amount with SI = P + SI.
• CI amount after n years = P(1 + r/100)^n.
• For two years, CI − SI = P(r/100)^2.

Time and work

• If A finishes a job in x days, one-day work = 1/x.
• Together, add work rates: 1/x + 1/y.
• Time = 1/(combined rate).
• If workers and days have equal efficiency, workers multiplied by days is constant.
• Pipes: inlet rates are positive; outlet/leak rates are negative.

Speed, distance and time

• Distance = speed multiplied by time.
• Speed = distance/time.
• Average speed for equal distances at speeds x and y = 2xy/(x+y).
• Convert km/h to m/s: multiply by 5/18.
• Convert m/s to km/h: multiply by 18/5.
• Relative speed: add for opposite directions; subtract for same direction.
• Train passing a pole: train length/speed.
• Train passing a platform: (train length + platform length)/speed.

Mixtures

Use alligation for two quantities:

quantity of cheaper : quantity of dearer = (dearer price − mean price) : (mean price − cheaper price).

For concentration, write the amount of pure substance, not only the total mixture.

Number system

• For two numbers: HCF multiplied by LCM = product, only when the two numbers are positive integers.
• Divisibility: 2 last digit even; 3 digit sum divisible by 3; 4 last two digits divisible by 4; 5 last digit 0/5; 9 digit sum divisible by 9; 11 alternating digit sum divisible by 11.
• Remainder after division: keep reducing at each arithmetic step.
• Unit digit cycles: 2 → 2,4,8,6; 3 → 3,9,7,1; 4 → 4,6; 7 → 7,9,3,1; 8 → 8,4,2,6; 9 → 9,1.

Permutation, combination and probability

• n! = n(n−1)(n−2)...1.
• Permutations: nPr = n!/(n−r)!.
• Combinations: nCr = n!/[r!(n−r)!].
• Probability = favourable outcomes/total equally likely outcomes.
• "At least one" is often easier as 1 − probability(none).

[[02_Aptitude/Aptitude_Overview|Aptitude overview]] · [[09_Revision/Last_Hour_Cheat_Sheet|Last-hour cheat sheet]]