TIME
AND DISTANCE -> IMPORTANT FACTS AND FORMULAE
1.
Speed = [Distance/Time],
Time=[Distance/Speed],
Distance
= (Speed*Time)
2.
x km/hr = [x*5/18] m/sec.
3.
If the ratio of the speeds of A and B is a:b, then the ratio of the times taken
by them to cover the same distance is 1/a : 1/b or b:a.
4.
x m/sec = [x*18/5] km/hr.
5.
Suppose a man covers a certain distance at x km/hr and an equal distance at y
km/hr. then, the average speed during the whole journey is [2xy/x+y] km/hr.
PROFIT
AND LOSS -> IMPORTANT FACTS AND FORMULAE
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.
1.
Gain = (S.P.) - (C.P.)
2.
Loss or gain is always reckoned on C.P.
3.
gain% = [Gain*100/C.P.]
4.
Loss = (C.P.) - (S.P.)
5.
Loss% = [Loss*100/C.P.]
6.
S.P. = (100+Gain%)/100 * C.P.
7.
S.P. = (100-Loss%)/100 * C.P.
8.
C.P. = 100/(100+Gain%) * S.P.
9.
C.P. = 100/(100-Loss%) * S.P.
10.
If an article is sold at a gain of say, 35%, then S.P. = 135% of CP
11.
If an article is sold at a loss of say, 35%, then S.P. = 65% of C.P.
VOLUME AND
SURFACE AREA -> IMPORTANT FACTS AND FORMULAE
I. CUBIOD
Let length =
l, breadth = b and height = h units. Then,
1. Volume =
(l x b x h) cubic units.
2. Surface
area = 2 (lb + bh + lh)
II. CUBE
Let each
edge of a cube be of length a. Then, 1. Volume = a³ cubic units.
2. Surface
area = 6a² sq. units.
3. Diagonal
= √3 a units.
III.
CYLINDER
Let radius
of base = r and Height (or length) = h Then,
1. Volume =
(Πr²h) cubic units.
2. Curved
surface area = (2Πrh) sq. units.
3. Total
surface area = (2Πrh + 2Πr² sq. units)
= 2Πr (h +
r) sq. units.
IV. CONE
Let radius
of base = r and Height = h. Then,
1. Slant
height, l = √h² + r ² units.
2. Volume =
[1/3 Πr²h] cubic units.
3. Total
surface area = (Πrl + Πr²) sq.units.
V. SPHERE
Let the
radius of the sphere be r. Then,
1. Volume =
[4/3 Πr3] cubic units.
2. Surface
area = (4Πr²) sq. units.
VI.
HEMISPHERE
Let the
radius of a hemisphere be r. Then,
1. Volume =
[2/3 Πr3] cubic units.
2. Curved
surface area = (3Πr²) sq. units.
3. Total
surface area = (3Πr²) sq. units.
Remember
: 1 litre = 1000 cm³.
BOATS AND
STREAMS -> IMPORTANT FACTS AND FORMULAE
I. In water,
the direction along the stream is called downstream. And, the direction against
the stream is called upstream.
II. 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.
III. If the
speed downstream is a km/hr and the speed upstream is b km/hr, then :
Speed in strill
water = 1/2 (a + b) km/hr
Rate of stream = 1/2 (a - b) km/hr
PARTNERSHIP
-> IMPORTANT FACTS AND FORMULAE
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 -> IMPORTANT CONCEPTS
Bankers’
Discount : Suppose a merchant A buys googds 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 Dicount (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.
BANKERS
DISCOUNT -> IMPORTANT FORMULAE
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]
CLOCKS ->
IMPORTANT FORMULAE
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 onece.
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 munute 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.
TRUE
DISCOUNT -> IMPORTANT CONCEPTS
Suppose a
man has to pay Rs. 156 after 4 years and the rate of interest is 14% per annum.
Clearly, Rs. 100 at 14% will amount to Rs. 156 in 4 years. So, the payment of
Rs. 100 now will clear off the debt of Rs. 156 due 4 years hence. We say that :
Sum
due = Rs. 156 due 4 years hence;
Present
worth (P.W.) = Rs.100;
True
Discount (T.D.) = Rs. (156 - 100) = Rs. 56 = (Sum due) - (P.W.).
We define :
T.D. = Interest on P.W.
Amount =
(P.W.) + (T.D.).
Interest is
reckoned on P.W. and true discount is reckoned on the amount.
TRUE
DISCOUNT -> IMPORTANT FORMULAE
Let rate =
R% per annum 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 -> IMPORTANT FORMULAE
1. a km/hr =
[a * 5/18]m/s.
2. a m/s =
[a * 18/5] km/hr.
3. Time
taken by a trian of length l metres to pass a pole or a standing man or a
signal post is equal to the time taken by the train to cover l metres.
4. Time
taken by a train of length l metres to pass a stationary object of length b
metres is the time taken by the train to cover (l + b) metres.
5. 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.
6. 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
7. If two
trains of length a metres and b metres are moving in opposite directions at u
8. If two
trains of length a metres and b metres are moving in the same direciton 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.
9. 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).
SIMPLE
INTEREST -> IMPORTANT FORMULAE
1. Principal
: The money borrowed or lent out for a certain priod is called the principal of
he sum.
2. Interest
: Extra money paid for using other’s money is called interest.
3. 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 annum (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 -> DESCRIPTION
In
this section, questions involving a set of numbers are put in the form of a
puzzle. You have to analyse the given conditions, assume the unknown the
numbers and form equations accordingly, which on solving yield the unknown
numbers.
AVERAGE
-> IMPORTANT FACTS AND FORMULAE
I. Average =
[Sum of observations / Number of observations]
II. Suppose
a man covers a certain distance at x kmph and an equal distance at y kmph.
Then, the average speed during the whole journey is [2xy / x + y] kmph.
Numbers
-> IMPORTANT FACTS AND FORMULAE
1. 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 Formulae :
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 -> IMPORTANT
FACTS AND FORMULAE
1. 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
2. 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
