Formulae for Streams and Boats

1. Downstream/Upstream:
   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/hr 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	(a + b) km/hr.
   	2

   Rate of stream =	1	(a - b) km/hr.
   	2

Formulae for Ration and Proportion

1. Ratio:
   The ratio of two quantities a and b in the same units, is the fraction  and we write it as a : b.
   In the ratio a : b, we call a as the first term or Antecedent and b, the second term or Consequent.

   Eg. The ratio 5 : 9 represents	5	with antecedent = 5, consequent = 9.
   	9

   Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
   Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

1. Proportion:
   The equality of two ratios is called proportion.
   If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
   Here a and d are called extremes, while b and c are called mean terms.
   Product of means = Product of extremes.
   Thus, a : b :: c : d  (b x c) = (a x d).

1. Fourth Proportional:
   If a : b = c : d, then d is called the fourth proportional to a, b, c.
   Third Proportional:
   a : b = c : d, then c is called the third proportion to a and b.
   Mean Proportional:
   Mean proportional between a and b is ab.

1. Comparison of Ratios:

   We say that (a : b) > (c : d)	a	>	c	.
   	b		d

   Compounded Ratio:
   The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

1. Duplicate Ratios:
   Duplicate ratio of (a : b) is (a² : b²).
   Sub-duplicate ratio of (a : b) is (a : b).
   Triplicate ratio of (a : b) is (a³ : b³).
   Sub-triplicate ratio of (a : b) is (a^1/3 : b^1/3).

   If	a	=	c	, then	a + b	=	c + d	.     [componendo and dividendo]
   	b		d		a - b		c - d

2. Variations:
   We say that x is directly proportional to y, if x = ky for some constant k and we write,x  y.
   We say that x is inversely proportional to y, if xy = k for some constant k and

   we write, x	1	.
   	y

Formulae on Volume and Surface Area

1. CUBOID
   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) sq. units.
   3. Diagonal = l² + b² + h² units.
2. 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 = 3a units.
3. CYLINDER
   Let radius of base = r and Height (or length) = h. Then,
   1. Volume = (r²h) cubic units.
   2. Curved surface area = (2rh) sq. units.
   3. Total surface area = 2r(h + r) sq. units.
4. CONE
   Let radius of base = r and Height = h. Then,
   1. Slant height, l = h² + r² units.
   2. Volume = r²h cubic units.
   3. Curved surface area = (rl) sq. units.
   4. Total surface area = (rl + r²) sq. units.
5. SPHERE
   Let the radius of the sphere be r. Then,
   1. Volume = r³ cubic units.
   2. Surface area = (4r²) sq. units.
6. HEMISPHERE
   Let the radius of a hemisphere be r. Then,
   1. Volume = r³ cubic units.
   2. Curved surface area = (2r²) sq. units.
   3. Total surface area = (3r²) sq. units.
      Note: 1 litre = 1000 cm³.

Basics of Clock

1. Minute Spaces:
   The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.
   Hour Hand and Minute Hand:
   A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand.
2. 1. In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.
   2. In every hour, both the hands coincide once.
   3. The hands are in the same straight line when they are coincident or opposite to each other.
   4. When the two hands are at right angles, they are 15 minute spaces apart.
   5. When the hands are in opposite directions, they are 30 minute spaces apart.
   6. Angle traced by hour hand in 12 hrs = 360°
   7. Angle traced by minute hand in 60 min. = 360°.
   8. 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.

Basics of Percentage

1. Concept of Percentage:
   By a certain percent, we mean that many hundredths.
   Thus, x percent means x hundredths, written as x%.

   To express x% as a fraction: We have, x% =	x	.
   	100

   Thus, 20% =	20	=	1	.
   	100		5

   To express	a	as a percent: We have,	a	=		a	x 100	%.
   	b		b			b

   Thus,	1	=		1	x 100	%	= 25%.
   	4			4

2. Percentage Increase/Decrease:
   If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is:

   	R	x 100	%
   	(100 + R)

   If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is:

   	R	x 100	%
   	(100 - R)
3. Results on Population:
   Let the population of a town be P now and suppose it increases at the rate of R% per annum, then:

   1. Population after n years = P		1 +	R		n
   			100

   2. Population n years ago =	P
   	1 + R n 100

4. Results on Depreciation:
   Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:

   1. Value of the machine after n years = P		1 -	R		n
   			100

   2. Value of the machine n years ago =	P
   	1 - R n 100

   3. If A is R% more than B, then B is less than A by		R	x 100	%.
   		(100 + R)

   4. If A is R% less than B, then B is more than A by		R	x 100	%.
   		(100 - R)

Formulae on Compound Interest

1. Let Principal = P, Rate = R% per annum, Time = n years.
2. When interest is compound Annually:

   Amount = P		1 +	R		n
   			100

3. When interest is compounded Half-yearly:

   Amount = P		1 +	(R/2)		2n
   			100

4. When interest is compounded Quarterly:

   Amount = P		1 +	(R/4)		4n
   			100

5. When interest is compounded Annually but time is in fraction, say 3 years.

   Amount = P		1 +	R		3	x		1 +	R
   			100						100

6. When Rates are different for different years, say R₁%, R₂%, R₃% for 1^st, 2^nd and 3^rd year respectively.

   Then, Amount = P

   1 +

   R1

   1 +

   R2

   1 +

   R3

   .

   100

   100

   100

7. Present worth of Rs. x due n years hence is given by:

   Present Worth =	x	.
   	1 + R 100