Aptitude Shortcut Formulas
Posted by admin at June 20, 2019
SPEED, TIMES AND DISTANCE
Concept of speed, time and distance is based on the formula
Speed × time=Distance
The rate at which any moving body covers a particular distance is called is speed.
It is the time duration over which the movement has occurred. The unit used for measuring time is synchronous with denominator of the unit used for measuring speed. Thus, if the speed is measured in terms of km/h then time is measured in hours’
SI unit of speed is metre per second (mps). It is also measured in kilometres per hour (kmph) or miles per hour (mph).
Conversion of units:
- 1 hour=60 minutes=60×60 seconds
- 1km=1000 m
- 1km=0.625 mile
- 1 mile=1.60 km, i.e. 8km=5 miles
- 1 yard=3 feet
- 1 km/h=5/18m/sec,
- 1 m/sec= 18/5 km/h
- 1 miles/hr = 22/15ft/sec.
- Average speed= (Total distance)/(Total time)
- While travelling a certain distance d, if a man changes his speed in the ration m:n, then the ratio of time taken becomes n:m
- If a certain distance(d), say from A to B, is covered at ‘a’ km/hr and the same distance is covered again say from B to A in ‘b’ km/hr, then the average speed during the wholesomely is given by: Average speed= [2ab/(a+b)]km/h…(which is the harmonic means of a and b
PROFIT AND LOSS
Cost Price : The price at which an article is purchased, is called its cost price, abbreviated as C.P.
Selling Price : The price at which an article is purchased, is called its cost price, abbreviated as C.P.
Profit or Gain : The price at which an article is purchased, is called its cost price, abbreviated as C.P.
Loss : If S.Pis less than C.P., the seller is said to have incurred a loss.
- Gain = (S.P.) – (C.P.)
- Loss or gain is always reckoned on C.P.
- gain% = [Gain*100/C.P.]
- Loss = (C.P.) – (S.P.)
- Loss% = [Loss*100/C.P.]
- S.P. = (100+Gain%)/100 * C.P.
- S.P. = (100-Loss%)/100 * C.P.
- C.P. = 100/(100+Gain%) * S.P.
- C.P. = 100/(100-Loss%) * S.P.
VOLUME AND SURFACE AREA
Let length = l, breadth = b and height = h units. Then,
- Volume = (l x b x h) cubic units.
- Surface area = 2 (lb + bh + lh)
Let each edge of a cube be of length a. Then, 1. Volume = a³ cubic units.
- Surface area = 6a² sq. units.
- Diagonal = √3 a units.
Let radius of base = r and Height (or length) = h Then,
- Volume = (Πr²h) cubic units.
- Curved surface area = (2Πrh) sq. units.
- Total surface area = (2Πrh + 2Πr² sq. units)
= 2Πr (h + r) sq. units.
Let radius of base = r and Height = h. Then,
- Slant height, l = √h² + r ² units.
- Volume = [1/3 Πr²h] cubic units.
- Total surface area = (Πrl + Πr²) sq.units.
Let the radius of the sphere be r. Then,
- Volume = [4/3 Πr3] cubic units.
- Surface area = (4Πr²) sq. units.
Let the radius of a hemisphere be r. Then,
- Volume = [2/3 Πr3] cubic units.
- Curved surface area = (3Πr²) sq. units.
- Total surface area = (3Πr²) sq. units.
Remember : 1 litre = 1000 cm³.
BOATS AND STREAMS
1. In water, the direction along the stream is called downstream. And, the direction against the stream is called upstream.
2. If the speed of a boat in still water is u km/ht and the speed of the stream is v km/hr, then :
Speed downstream = (u + v) km/hr
Speed upstream (u – v) km/hr.
3. If the speed downstream is a km/hr and the speed upstream is b km/hr, then :
Speed in still water = 1/2 (a + b) km/hr
Rate of stream = 1/2 (a – b) km/hr
I. Partnership : When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.
II. Ratio of Division of Gains :
(i) When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.
Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year :
(A’s share of profit) : (B’s share of profit) = x : y.
(ii) When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital * number of units of time). Now, gain or loss is divided in the ratio of these capitals.
Suppose A invests Rs. x for p months and B invests Rs. y for q months, then (A’s share of profit) : (B’s share of profit) = xp : yq.
III. Working and Sleeping Partners : A partner who manages the business is known as working partner and the one who simply invests the money is a sleeping partner.
Bankers’ Discount : Suppose a merchant A buys goods worth, say Rs. 10,000 from another merchant B at a credit of say 5 months. Then, B prepares a bill, called the bill of exchange. A signs this bill and allows B to withdraw the amount from his bank account after exactly 5 months.
The date exactly after 5 months is called nominally due date. Three days (known as grace days) are added to it to get a date, known as legally due date.
Suppose B wants to have the money before the legally due date. Then he can have the money from the banker or a broker, who deducts S.I. on the face value (i.e., Rs. 10,000 in this case) for the period from the date on which the bill was discounted (i.e., paid by the banker) and the legally due date. This amount is known as Banker’s Discount (B.D.)
Thus, B.D. is the S.I. on the face value for the period from the date on which the bill was discounted and the legally due date.
Banker’s Gain (B.G.) = (B.D.) – (T.D.) for the unexpired time.
Note : When the date of the bill is not given, grace days are not to be added.
I. B.D. = S.I. on bill for unexpired time.
II. B.G. = (B.D.) – (T.D.) = S.I. on T.D. = (T.D.)² / R.W.
III. T.D. = √P.W. * B.G.
IV. B.D. = [Amount * Rate * Time / 100]
V. T.D. = [Amount * Rate * Time / 100 + (Rate * Time)]
VI. Amount = [B.D. * T.D. / B.D. – T.D.]
VII. T.D. = [B.G. * 100 / Rate * Time]
The face or dial of a watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.
A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called the minute hand or long hand.
I. In 60 minutes, the minute hand gains 55 minutes on the hour hand.
II. In every hour, both the hands coincide once.
III. The hands are in the same straight line when they are coincident or opposite to each other.
IV. When the two hands are at right angles, they are 15 minute spaces apart.
V. When the hands are in opposite directions, they are are 30 minute spaces apart.
VI. Angle traced by hour hand in 12 hrs = 360°.
VII. Angle traced by minute hand in 60 min. = 360°.
Too Fast and Too Slow : If a watch or a clock indicates 8.15, when the correct time is 8, it is said to be 15 minutes too fast.
On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.
Let rate = R% per annul and Time = T years. Then,
I. P.W. = 100 * Amount / 100 + (R*T) = 100 * T.D. / R * T
II. T.D. = (P.W.)* R * T / 100 = Amount * R * T / 100 + (R * T)
III. Sum = (S.I.) * (T.D.) / (S.I.) – (T.D.)
IV. (S.I.) – (T.D.) = S.I on T.D.
V. When the sum is put at compound interest, then P.W. = Amount / [1+R/100]T;
PROBLEMS ON TRAINS
- a km/hr = [a * 5/18]m/s.
- a m/s = [a * 18/5] km/hr.
- Time taken by a train of length l meters to pass a pole or a standing man or a signal post is equal to the time taken by the train to cover l meters.
- Time taken by a train of length l meters to pass a stationary object of length b meters is the time taken by the train to cover (l + b) meters.
- Suppose two trains or two bodies are moving in the same direction at u m/s and v m/s, where u>v, then their relatives speed = (u – v) m/s.
- Suppose two trains or two bodies are moving in opposite directions at u m/s and v m/s, then their relative speed is = (u + v) m/s
- If two trains of length a meters and b meters are moving in opposite directions at u
- If two trains of length a meters and b meters are moving in the same direction at u m/s and v m/s, then the time taken by the faster train to cross the slower train = (a + b)/(u – v) sec.
- If tow trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then
(A’s speed) : (B’s speed) = (√b : √a).
- Principal : The money borrowed or lent out for a certain period is called the principal of he sum.
- Interest : Extra money paid for using other’s money is called interest.
- Simple Interest (S.I.) : If the interest on a sum borrowed for a certain period is reckoned uniformly, then it is called simple interest.
Let Principal = P, Rate = R% per annul (p.a.) and Time = T years, Then,
(i) S.I. = [P * R * T / 100]
(ii) P = [100 * S.I. / R * T]
R = [100 * S.I / P * T] and T = [100 * S.I. / P * R]
PROBLEMS ON NUMBERS
I. Average = [Sum of observations / Number of observations]
II. Suppose a man covers a certain distance at x kmph and an equal d distance at y kmph. Then, the average speed during the whole journey is [2xy / x + y] kmph.
Numbers -> IMPORTANT FACTS AND FORMULAE
- Natural Numbers :
Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.
II. Whole Numbers :
All counting numbers together with zero form the set of whole numbers. Thus,
I. 0 is the only whole number which is not a natural number.
II. Every natural number is a whole number.
III.Some Important Formulas :
I. ( 1 + 2 + 3 + …..+ n) = n (n + 1 ) / 2
II. (1 2 + 22 + 32 + ….. + n2) = n ( n + 1 ) (2n + 1) / 6
III. (1 3 + 23 + 33 + ….. + n3) = n2 (n + 1)2 / 4
SURDS ADN INDICES
- LAWS OF INDICES :
(i) am * an = am + n
(ii) am / an = am – n
(iii) (am)n = amn
(iv) (ab)n = anbn
(v) (a/b)n = an/ bn
(vi) a0 = 1
- SURDS : Let a be rational number and n be a positive integer such
that a(1/n) = n√a
3 LAWS OF SURDS :
(i) n√a = a (1/n)
(ii) n√ab = n√a x n√b
(iii) n√a/b = n√a / n√b
(iv) (n√a)n = a