Mental Ability Formulas and Shortcuts — Complete One-Page Reference

Mental ability questions carry 10-15 marks in every PSC paper. This one-page formula sheet covers all topics with shortcuts for fast solving.

1. Percentage

Formula	Expression
Percentage	(Part / Whole) x 100
A is what % of B	(A / B) x 100
A is how much % more than B	((A - B) / B) x 100
A is how much % less than B	((B - A) / B) x 100
If price increases by x%, to restore original: decrease by	(x / (100 + x)) x 100
If price decreases by x%, to restore original: increase by	(x / (100 - x)) x 100
Successive % change: x% then y%	Net = x + y + (xy/100)

Fraction-Percentage Quick Conversions:

Fraction	%	Fraction	%
1/2	50%	1/8	12.5%
1/3	33.33%	1/9	11.11%
1/4	25%	1/10	10%
1/5	20%	1/11	9.09%
1/6	16.67%	1/12	8.33%
1/7	14.28%	2/3	66.67%

2. Profit and Loss

Formula	Expression
Profit	SP - CP
Loss	CP - SP
Profit %	(Profit / CP) x 100
Loss %	(Loss / CP) x 100
SP = CP x (100 + P%) / 100	When profit
SP = CP x (100 - L%) / 100	When loss
CP = SP x 100 / (100 + P%)	Finding CP from SP (profit case)
If marked price and discount: SP	MP x (100 - D%) / 100
Two articles sold at same SP, one at x% profit and other at x% loss	Net loss always = x²/100 %
Dishonest dealer using false weight	Gain% = (True weight - False weight) / False weight x 100

3. Simple Interest (SI)

Formula	Expression
SI	(P x R x T) / 100
Amount (A)	P + SI = P(1 + RT/100)
Principal	(SI x 100) / (R x T)
Rate	(SI x 100) / (P x T)
Time	(SI x 100) / (P x R)
If SI = Principal	T = 100/R years
If amount doubles	T = 100/R years
If amount triples	T = 200/R years
If amount becomes n times	T = (n-1) x 100/R years

4. Compound Interest (CI)

Formula	Expression
Amount	A = P(1 + R/100)^T
CI	A - P = P[(1 + R/100)^T - 1]
CI for 2 years vs SI	Difference = P x (R/100)²
CI for 3 years vs SI	Difference = P x (R²(300+R)) / 100³
Half-yearly compounding	A = P(1 + R/200)^(2T)
Quarterly compounding	A = P(1 + R/400)^(4T)

Quick CI Shortcuts (2-year):

Rate	Effective rate for 2 years
5%	10.25%
10%	21%
15%	32.25%
20%	44%

Formula: Effective rate for 2 years = 2R + R²/100

5. Time and Work

Formula	Expression
If A does work in ‘a’ days, A’s 1 day work	1/a
If A and B work together	1/a + 1/b = (a+b)/ab; Total time = ab/(a+b)
If A, B, C work together	Time = 1/(1/a + 1/b + 1/c)
If A and B together in x days, A alone in a days, B alone in	1/b = 1/x - 1/a; b = ax/(a-x)
Pipes: fill rate - drain rate	1/a - 1/b (if pipe A fills, B drains)
If M1 men do W1 work in D1 days and M2 men do W2 work in D2 days	M1 x D1 / W1 = M2 x D2 / W2
Including hours: M1D1H1/W1 = M2D2H2/W2	MDH/W is constant

Shortcut: If A is x times as efficient as B, and A takes ‘a’ days, then B takes ‘xa’ days. Together: xa/(x+1) days.

6. Speed, Distance, and Time

Formula	Expression
Distance	Speed x Time
Speed	Distance / Time
Time	Distance / Speed
km/h to m/s	Multiply by 5/18
m/s to km/h	Multiply by 18/5
Average speed (same distance, two speeds)	2S1S2 / (S1 + S2)
Average speed (same time, two speeds)	(S1 + S2) / 2
Relative speed (same direction)	S1 - S2
Relative speed (opposite direction)	S1 + S2
Train crossing a pole	Time = Length of train / Speed
Train crossing a platform	Time = (Length of train + Length of platform) / Speed
Two trains crossing each other	Time = (L1 + L2) / Relative speed

7. Averages

Formula	Expression
Average	Sum of values / Number of values
Sum	Average x Number
If one number is added	New avg = (Old sum + new number) / (n + 1)
If one number is removed	New avg = (Old sum - removed number) / (n - 1)
If one number is replaced	Change in sum = New number - Old number
Weighted average	(w1.x1 + w2.x2 + …) / (w1 + w2 + …)
Average of first n natural numbers	(n + 1) / 2
Average of first n even numbers	(n + 1)
Average of first n odd numbers	n
Sum of first n natural numbers	n(n + 1) / 2
Sum of squares of first n numbers	n(n+1)(2n+1) / 6
Sum of cubes of first n numbers	[n(n+1)/2]²

8. Ratio and Proportion

Formula	Expression
a : b = c : d means	a/b = c/d (or ad = bc)
Componendo	(a+b)/b = (c+d)/d
Dividendo	(a-b)/b = (c-d)/d
If a:b = 2:3 and b:c = 4:5	Make b common: a:b:c = 8:12:15
Dividing X in ratio a:b	Parts are Xa/(a+b) and Xb/(a+b)
Mixture: to find ratio for mixing at prices p1 and p2 to get avg price m	(p2-m) : (m-p1) — Alligation rule

9. Number Series

Pattern	Example
Arithmetic (constant difference)	2, 5, 8, 11, 14 (d = +3)
Geometric (constant ratio)	3, 6, 12, 24, 48 (r = x2)
Difference of differences	1, 2, 5, 10, 17 (diff: 1, 3, 5, 7 — AP)
Squares	1, 4, 9, 16, 25, 36
Cubes	1, 8, 27, 64, 125
Fibonacci-type	1, 1, 2, 3, 5, 8, 13
Alternating operations	+2, x3, +2, x3…
Prime numbers	2, 3, 5, 7, 11, 13, 17, 19, 23, 29

10. Age Problems

Shortcut	Formula
If A is x years older than B	A - B = x (constant, always)
Present ratio a:b, after t years ratio c:d	Let ages = ax, bx; solve (ax+t)/(bx+t) = c/d
If sum of ages = S and ratio = a:b	Ages = Sa/(a+b) and Sb/(a+b)

11. Calendar

Fact	Detail
Odd days in 1 ordinary year	1 (365 = 52 weeks + 1 day)
Odd days in 1 leap year	2 (366 = 52 weeks + 2 days)
Odd days in 100 years	5
Odd days in 200 years	3
Odd days in 300 years	1
Odd days in 400 years	0
Day codes	Sun=0, Mon=1, Tue=2, Wed=3, Thu=4, Fri=5, Sat=6
Leap year rule	Divisible by 4; century years by 400

12. Clock Problems

Formula	Expression
Angle between hands	abs(30H - 5.5M) degrees (H = hour, M = minutes)
Hands coincide	Every 65 5/11 minutes
Hands opposite (180 degrees)	Every 65 5/11 minutes (offset from coincidence)
Hands at right angle (90 degrees)	22 times in 12 hours
Hands coincide in 12 hours	11 times
Hands opposite in 12 hours	11 times

13. Probability

Formula	Expression
P(Event)	Favourable outcomes / Total outcomes
P(not A)	1 - P(A)
P(A or B)	P(A) + P(B) - P(A and B)
P(A and B) if independent	P(A) x P(B)
Coin: P(Head)	1/2
Die: P(any number)	1/6
Cards: total	52 (4 suits x 13 each)

14. Permutation and Combination

Formula	Expression
nPr (permutation)	n! / (n-r)!
nCr (combination)	n! / [r!(n-r)!]
nC0 = nCn	1
nC1	n
nCr = nC(n-r)	Symmetry property
Circular permutation	(n-1)!