Knowledge Share: Simple Interest and Compound Interest formulas:: Compound Interest formulas: Let Principal = P, Rate = R% per annum, Time = n years.
When interest is compound Annually:
Amount = ...
Sunday 28 August 2011
Simple Interest and Compound Interest formulas:
- Compound Interest formulas:
Let Principal = P, Rate = R% per annum, Time = n years. - When interest is compound Annually:
Amount = P 
1 + R 
n 100 - When interest is compounded Half-yearly:
Amount = P 
1 + (R/2) 
2n 100 - When interest is compounded Quarterly:
Amount = P 1 + (R/4) 4n 100 - When interest is compounded Annually but time is in fraction, say 3
years.Amount = P 1 + R 3 x 
1 + R
100 100 - When Rates are different for different years, say R1%, R2%, R3% for 1st, 2ndand 3rd year respectively.
Then, Amount = P 1 + R1 1 + R2 1 + R3 . 100 100 100 - Present worth of Rs. x due n years hence is given by:
Present Worth = x . 1 + R 100
Simple Interest:
- Principal:The money borrowed or lent out for a certain period is called the principal or thesum.
- Interest:Extra money paid for using other's money is called interest.
- Simple Interest (S.I.):If the interest on a sum borrowed for 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). Simple Intereest = P x R x T 100
(ii). P = 100 x S.I. ; R = 100 x S.I. and T = 100 x S.I. . R x T P x T P x R
Tuesday 23 August 2011
Ratio and Proportion
Ratio and Proportion
- Compounded Ratio of two ratios a/b and c/d is ac/bd, i.e., ac : bd.
Duplicate ratio of a : b is a² : b²
Triplicate ratio of a : b is a³ : b³
Sub-duplicate ratio of a : b is
a : b
Sub-triplicate ratio of a : b is³a :³b
Reciprocal ratio of a : b is b : a - Proportion. Four (non-zero) quantities of the same kind a, b, c, d are in proportion,
written as a : b :: c : d iff a/b = c/d - The non-zero quantities of the same kind a, b, c, d, ... are in continued proportion iff
a/b = b/c = c/d = ...
In particular, a, b, c are in continued proportion iff a/b = c/d. In this case b is called the mean proportion; b =ac; c is called third proportional. If a, b, c, d are in proportion, then d is called fourth proportional. - Invertendo. If a : b :: c : d then b : a :: d : c
Alternendo. If a : b :: c : d then a : c :: b : d
Componendo. If a : b :: c : d then (a +b) : b :: (c +d) : d
Dividendo. If a : b :: c : d then (a -b) : b :: (c -d) : d
Componendo and dividendo.
If a : b :: c : d then (a +b) : (a -b) :: (c +d) : (c -d)
i.e., a/b = c/d => (a +b)/(a - b) = (c +d)/(c +d) - If a/b = c/d = e/f = ..., then each ratio = (a +c +e +...)/(b +d +f +...)
Exercise
- Find the ratio of
(i) 45 minutes to 5¾ hours
(ii) 4 months and 2½ years
(iii) 1·2 kg and 60 gm. - Find the compounded ratio of
(i) 5 : 7 and 9 : 10
(ii) (x +y) : (x -y) and (x -y) : (x +y)
(iii) 2a : 3b, 2b : 3a, a² : b² - Find the following
(i) the duplicate ratio of 3 : 7
(ii) the triplicate ratio of 2 : 5
(iii) the sub-duplicate ratio of 36 : 25
(iv) the sub-triplicate ratio of 27 : 1
(v) the reciprocal ratio of 9 : 11. - Find the following
(i) the duplicatae ratio of 2x : 3y
(ii) the sub-duplicatae ratio of 16a² : 25b²
(iii) the sub-triplicate ratio of a6 : 8b³ - Which ratio is greater 17 : 21 or 23 : 28?
- Arrange the following ratios in descending order of magnitude
13 : 9, 25 : 23, 16 : 11 and 20 : 17 - Arrange the ratios 2 : 3, 3 : 4, 4 : 5 in ascending order of magnitude.
- A man earns Rs 5000 per month and spends Rs 3500 per month. Find the ratio of his
(i) expenditure to income
(ii) savings to income
(iii) savings to expenditure. - If A : B = 4 : 5, B : C = 6 : 7, find A : C
- If A : B = 6 : 7, B : C = 8 : 9, find A : B : C
- Find the number which bears the same ratio to 7/33 that 8/21 bears to 4/9.
- Two numbers are in ratio 7 : 11. If 7 is added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.
- A ratio is equal to 3 : 7. If the antecedent is 5, find the consequent.
- Find two numbers in the ratio of 8 : 7 such that when each is decreased by 12½, they are in ratio 11 : 9.
- Divide Rs 170 in the ratio 3 : 7.
- Find the value of x in the following proportions
(i) 10 : 35 = x : 42
(ii) 3 : x = 24 : 2
(iii) 2·5 : 11·5 = x : 3
(iv) x : 50 :: 3 : 2. - Find the fourth proportional to
(i) 1/3, 1/4, 1/5
(ii) 1·5, 2·5, 4·5
(iii) 9·6 kg, 7·2 kg, 28·8 kg
(iv) 2xy, x², y² - Find the third proportional to
(i) 5, 10
(ii) 1·3, 3·9
(iii) 21/4 and 7
(iv) x² -y², (x +y)²
(v) 2 x, 4 x². - Find the mean proportion of
(i) 5 and 80
(ii) 1/12 and 1/75
(iii) 8·1 and 2·5
(iv)5 +2, 5 -2
(v) (a -b)³, (a -b)5 - What should be subtracted from 23, 30, 57 and 78 so that the remainders may be in proportion?
- What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
- Find two numbers such that their mean proportional is 9 and the third proportional is 243.
- If three quantities are in continued proportion, prove that first is to third is the duplicate ratio of the first to the second.
- If a
b and a : b is the duplicate ratio of (a +c) and (b +c), prove that c is the mean proportional between a and b. - If b is the mean proportional between a and c, prove that
(a² -b² +c²)/(a-2 -b-2 +c-2) = b4 - If x/a = y/b = z/c, prove that each ratio is equal to
- If y is the mean proportional between x and z, prove that xyz(x +y +z)³ = (xy +yz +zx)³.
- If x/a = y/b = z/c, prove that
(i)
(ii)
- If x/(b-c) = y/(c-a) = z/(a-b), prove that
(i) ax +by +cz = 0
(ii) x +y +z = 0 - If ax = by = cz, prove that x²/yz + y²/xz + z²/xy = bc/a² + ca/b² +ab/c²
- Find x from the following equations:
(i) [(a +x) +(a -x)]/[(a + x) -(a - x)] = c/d
(ii) 16
Answers
1. (i) 3 : 23 (ii) 2 : 15 (iii) 20 : 12. (i) 9 : 14 (ii) 1 : 1 (iii) 4a² : 9b²
3. (i) 9 : 49 (ii) 9 : 125 (iii) 6 : 5 iv) 3 : 1 (v) 11 : 9
4. (i) 4x : 9y (ii) 4a : 5b (iii) a² : 2b. 5. 23 : 28
6. 16 : 11, 13 : 9, 20 : 17, 25 : 23 7. 2 : 3, 3 : 4, 4 : 5
8. (i) 7 : 10 (ii) 3 : 10 (iii) 3 : 7 9. 24 : 25. 10. 48 : 56 : 63
11. 2/11 12. 49 and 77 13. 35/3 14. 40, 35
15. Rs 51, Rs 119 16 (i) 12 (ii) 1/4 (iii) 5 (iv) 75
17. (i) 3/20 (ii) 7·5 (iii) 21·6 kg (iv) x y/2
18. (i) 20 (ii) 11·7 (iii) 28/3 (iv) (x +y)³/(x-y) (v) 8 x³
19. (i) 20 (ii) 1/30 (iii) 4·5 (iv)
3 (v) (a -b)420. 6 21. 3 22. 3,27
31. (i) 2acd/(c² +d²) (ii) a/3
Knowledge Share: Some good lines
Knowledge Share: Some good lines: Zamane Ki Har Jannat Aapke Liye Hogi, Yeh Aasma Ye Zami Aapke Liye Hogi, Mujhse Keh Bhi Na Paonge Uske Pehle Mere Hisse Ki Har Khushi A...
Some good lines
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Monday 22 August 2011
Knowledge Share: A Short Story
Knowledge Share: A Short Story: This story is almost a hundred years old. Seven-year old Bhim went to Goregaon in Maharashtra with his father to spend his holidays. He sa...
A Short Story
This story is almost a hundred years old. Seven-year old Bhim went to Goregaon in Maharashtra with his father to spend his holidays.
He saw a barber cutting the long hair of a rich farmer’s buffalo. He thought of his own long hair. He went to the barber and asked for a hair cut.
The barber replied, “If I cut your hair both my razor and I will get dirty.” Oh, so to cut human hair can be dirtier than cutting an animal’s hair, wondered little Bhim.
Later this little Bhim was known as Bhim Rao Baba Saheb Ambedkar. He became very famous across the world. Baba Saheb fought for justice for people like him.
After India’s freedom the Constitution was prepared under the leadership of Baba Saheb.
Knowledge Share: Long live INDIA ...
Knowledge Share: Long live INDIA ...: Long time ago in the great country called bharat there was a system in which brahmins were considered to be the persons who are going to te...
Long live INDIA ...
Long time ago in the great country called bharat there was a system in which brahmins were considered to be the persons who are going to teach and learn the vedas and deliever the goods to the society.But there was a catch , in return they were not going to get wealth so for livelihood they were to go to every household for Bhiksha which slowly and steadily termed as Bhikh and brahmins were considered as Beggars.
Today also if we see in the society somebody studying vedas and performing devotional duties and wearing saffron cloth considered as fakirs.Brahmins were getting poorer and poorer and for livelyhood they only had prasad to eat ...
That time we needed baba saheb for the upliftment of a particular cast and he is successful to an extent ...now we again need another baba saheb for amending the present constitution as it has become very old and needs revisions.
Today we are not a civilization with multiple cast , but we are a mixed culture with all the cultures working together forgetting who our last name signifies.. Today the core issues are corruption, reservation,inflation and vast gap between rich and poor.
There is a doubt in the society and even Baba saheb would have thoght of amending the 60 year old book we called The Constitution of India.
Long live INDIA ...
Knowledge Share: Anna
Knowledge Share: Anna: 10 things to know about Anna Hazare 'n Jan Lok Pal Bill.. ! 1. Who is Anna Hazare? An ex-army man. Fought 1965 Indo-Pak War 2. What...
Anna
10 things to know about Anna Hazare 'n Jan Lok Pal Bill.. !
1. Who is Anna Hazare?
An ex-army man. Fought 1965 Indo-Pak War
2. What's so special about him?
He built a village Ralegaon Siddhi in Ahamad Nagar district, Maharashtra
3. So what?
This village is a self-sustained model village. Energy is produced in the village itself from solar power, biofuel and wind mills.
In 1975, it used to be a poverty clad village. Now it is one of the richest village in India. It has become a model for self-sustained, eco-friendly & harmonic village.
4. Ok,...?
This guy, Anna Hazare was awarded Padma Bhushan and is a known figure for his social activities.
5. Really, what is he fighting for?
He is supporting a cause, the amendment of a law to curb corruption in India.
6. How that can be possible?
He is advocating for a Bil, The Jan Lokpal Bill (The Citizen Ombudsman Bill), that will form an autonomous authority who will make politicians (ministers), beurocrats (IAS/IPS) accountable for their deeds.
8. It's an entirely new thing right..?
In 1972, the bill was proposed by then Law minister Mr. Shanti Bhushan. Since then it has been neglected by the politicians and some are trying to change the bill to suit thier theft (corruption).
7. Oh.. He is going on a hunger strike for that whole thing of passing a Bill ! How can that be possible in such a short span of time?
The first thing he is asking for is: the government should come forward and announce that the bill is going to be passed.
Next, they make a joint committee to DRAFT the JAN LOKPAL BILL. 50% goverment participation and 50% public participation. Because you cant trust the government entirely for making such a bill which does not suit them.
8. Fine, What will happen when this bill is passed?
A LokPal will be appointed at the centre. He will have an autonomous charge, say like the Election Commission of India. In each and every state, Lokayukta will be appointed. The job is to bring all alleged party to trial in case of corruptions within 1 year. Within 2 years, the guilty will be punished. Not like, Bofors scam or Bhopal Gas Tragedy case, that has been going for last 25 years without any result.
9. Is he alone? Whoelse is there in the fight with Anna Hazare?
Baba Ramdev, Ex. IPS Kiran Bedi, Social Activist Swami Agnivesh, RTI activist Arvind Kejriwal and many more.
Prominent personalities like Aamir Khan is supporting his cause.
10. Ok, got it. What can I do?
At least we can spread the message.
Sunday 21 August 2011
Knowledge Share: Newton History
Knowledge Share: Newton History: Tradition has it that Newton was sitting under an apple tree when an apple fell on his head, and this made him understand that earthly and...
Newton History
Tradition has it that Newton was sitting under an apple tree when an apple fell on his head, and this made him understand that earthly and celestial gravitation are the same. A contemporary writer, William Stukeley, recorded in his Memoirs of Sir Isaac Newton’s Life a conversation with Newton in Kensington on April 15, 1726, in which Newton recalled “when formerly, the notion of gravitation came into his mind. It was occasioned by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth’s centre.”
Sir Isaac Newton, English mathematician, philosopher, and physicist, was born in 1642 in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. His father had died three months before Newton’s birth, and two years later his mother went to live with her new husband, leaving her son in the care of his grandmother. Newton was educated at Grantham Grammar School. In 1661 he joined Trinity College, Cambridge, and continued there as Lucasian professor of mathematics from 1669 to 1701. At that time the college’s teachings were based on those of Aristotle, but Newton preferred to read the more advanced ideas of modern philosophers such as Descartes, Galileo, Copernicus, and Kepler. In 1665, he discovered the binomial theorem and began to develop a mathematical theory that would later become calculus.
However, his most important discoveries were made during the two-year period from 1664 to 1666, when the university was closed due to the Great Plague. Newton retreated to his hometown and set to work on developing calculus, as well as advanced studies on optics and gravitation. It was at this time that he discovered the Law of Universal Gravitation and discovered that white light is composed of all the colors of the spectrum. These findings enabled him to make fundamental contributions to mathematics, astronomy, and theoretical and experimental physics.
Arguably, it is for Newton’s Laws of Motion that he is most revered. These are the three basic laws that govern the motion of material objects. Together, they gave rise to a general view of nature known as the clockwork universe. The laws are: (1) Every object moves in a straight line unless acted upon by a force. (2) The acceleration of an object is directly proportional to the net force exerted and inversely proportional to the object’s mass. (3) For every action, there is an equal and opposite reaction.
In 1687, Newton summarized his discoveries in terrestrial and celestial mechanics in his Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), one of the greatest milestones in the history of science. In this work he showed how his principle of universal gravitation provided an explanation both of falling bodies on the earth and of the motions of planets, comets, and other bodies in the heavens. The first part of the Principia, devoted to dynamics, includes Newton’s three laws of motion; the second part to fluid motion and other topics; and the third part to the system of the world, in which, among other things, he provides an explanation of Kepler’s laws of planetary motion.
This is not all of Newton’s groundbreaking work. In 1704, his discoveries in optics were presented in Opticks, in which he elaborated his theory that light is composed of corpuscles, or particles. Among his other accomplishments were his construction (1668) of a reflecting telescope and his anticipation of the calculus of variations, founded by Gottfried Leibniz and the Bernoullis. In later years, Newton considered mathematics and physics a recreation and turned much of his energy toward alchemy, theology, and history, particularly problems of chronology.
Newton achieved many honors over his years of service to the advancement of science and mathematics, as well as for his role as warden, then master, of the mint. He represented Cambridge University in Parliament, and was president of the Royal Society from 1703 until his death in 1727. Sir Isaac Newton was knighted in 1705 by Queen Anne. Newton never married, nor had any recorded children. He died in London and was buried in Westminster Abbey.
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