Hello Student`s this section is a very important part for those student those who are preparing for Competitive exams and they want to know the basic fundamental tips for Mathematics.
Most of the students got confused because of various units being used in different questions.
Examples:- A man goes at a speed of 20 miles per hour and another man goes at a speed of 33 kmph, who is moving faster. Solution:- 1 miles = 1600m = 1.6Km So,
20 miles = 20 x 1.6 = 32Km So a man goes at a speed of 33 Kmph is faster. Or, we can say conversion of units should be known.
Units used in time and distance:
1 hour
60min
1 mile
1600meters
1 Km
1000meters
1 mile
1760Yards
Note:-In case of distance or weight fundamental units are meter and gram. But various prefixes can be attached to the units. e.g., Kilo, hecta, deca, units, deci, centi, milli. All these prefixes has a difference of factor of 10.
1 Kilo = 10 Hecta = 102deca = 103Units = 104deci = 105Centi = 106milli. So we can convert one into another easily. Note:-If we want to convert km/h in m/sec then as 1 Km = 1000m and 1hr=3600sec. So,
1Km/h = 1000m/3600sec. 1 Km/hr = 5/18 m/sec In the questions of Data Interpretation we have Hindi as well as English numerical system. Terms defined
In Hindi
In English
1 thousand = 103
1 thousand = 103
1 lakh = 105
1 million = 106
1 Crore = 107
1 billion = 109
1 Arab = 109
1 trillion = 1012
1 Kharab = 1011
Other general terms used:
Dozen means 12
Score means 20
Hectare- unit of area means 10000m2
1m2 = 10000cm2
1m3 = 1000000cm3
RATIO AND PROPORTION
Ratio is the relation which one quantity bears to another of the same kind. The ratio of A to B is usually written A : B. The first term is called the antecedent and the second term is consequent.
a = ma b mb
i.e. the value of a ratio remains unaltered if the antecedent and the consequent are multiplied or divided by the same quantity. e.g.A and B have income in the ratio of 5 : 4 does not mean that the incomes are Rs.5 and Rs. 4, but it means any income which is a multiple of 5 and 4 , i.e 5x an 4x, where x is a variable.
Example 1:- Divide Rs 80 in the ratio of 3 : 5 ? Solution
Divide 80 by 8(sum of the ratios) we get 10 and have 10 is the k factor. Hence mutiply the ratio 3 : 5 by or 30 : 50.
Example 2:- Divide Rs. 98 among a,b,c,d in the ratio in the ratio 2:3:4:5? Solution. Divide Rs 98 by 14(sum of the ratios), we get 7 that is k factor. The amount is Rs. are 7 x 2,7 x 3, 7 x 4, 7 x 5 or 14, 21, 28, 35.
Ratios are compounded by multiplying together the fractions which denote them. Example 3:- Find the ratio compounded of the three ratios ? Solution:- 2a : 3b, 6ab : 5c2, c : a
Continued
When the ratio a : b is compounded with itself the resulting ratio is a2b2and is called the duplicate ratio of a : b.Also a1/2 : b1/2 is called the sub duplicate ratio of a : b. (i) The duplicate ratio of 2a : 3b is 4a2: 9b2 (ii) The sub duplicate ratio of 49 : 25 is 7:5
(iii) The triplicate ratio of 2x : 1 is 8x3 : 1
PROPORTION When two ratios are equal, the four quantities composing them are said to be proportionals. Thus if a/b = c/d them a, b, c, d are proportionals. This is expressed by saying that a is to b as c is to d, and the proportion is written a : b :: c : d. The terms a and d are called the extremes a and c the means
Let a, b, c, d be the proportionals. Then by definition a/b = c/d then ad = bc. Hence if any three terms of a proportion are given, the fourth may be find out. Thus if a, b, c,d are given, then b = ad/c
Quantities are said to be in continued proportion when the first is to the second, as the second is to the third If threequantities a, b, c, are in continued proportion , then a : b = b : c and ac = b2
If three quantities are proportional , the first is to the third is the duplicate ratio of the first to the second. Let three quantities be a, b, c, then a/b = b/c i.e a : c = a2 : b2 If a : b = c :d and e : f = g : h, then ae : bf = cg : dh
If four quantities a, b, c, d, form a proportion, many other proportions may be deduced by the properties of fractions.
if a : b = c : d, then b : a = d : c [ Invertendo ]
if a : b = c : d, then a : c = b : d [ Alternando ]
if a : b = c : d, then a+b : b = c+d : d [ Compenendo ]
if a : b = c : d, then a-b : b = c-d : d [ Dividendo ]
COMPARISON OF TWO RATIO Example:-If a : b = 2 :3 and b : c = 4 : 5, then what is the ratio of a : b : c ?
Solution
a = (2 x 4) = 8
b = (3 x 4) = 12
c = (3 x 5) = 15
So,the ratio a : b : c = 8 : 12 :15
Example:-What is the ratio of a, b, c, d. If the ratio of a : b = 2 : 3, b : c = 4 : 5 and c : d = 6 : 7?
Solution
a = 2 x 4 x 6 = 48
b = 3 x 4 x 6 = 72
c = 3 x 5 x 6 = 90
d = 3 x 5 x 7 = 105
So,the ratio a : b : c : d = 48 : 72 :90 : 105
FRACTIONAL RATIO Certain questions have to be solved according to their fractional ratio that means the ratio should be divided after solving fraction.
Example 1. A and B can complete a work in 2 & 3 days respectively. They complete a work together and get Rs. 50 as the wages. How should they divide it?
Solution:-Rs.50 is not being divided in the ratio of 2 : 3 as A who can complete the work in 2 days is more efficient then B who completed the work in 3 days.
Divide Rs. 50 in the ratio of So A should get Rs.30 and B should get Rs. 20
Example 2. A, B and C can complete a work in 2, 3 and 4 days respectively. If they complete the work together, in what ratio they should divide the money ?
Time and Distance
RELATIVE MOTION In this chapter the relative motion is most important concept. There is only one formula applicable
D = S x T
Some Important Conversion :
1 hour = 60minutes = 60 x 60 seconds
1 Kilometre = 1000metres
1 Kilometre = 0.6214 mile
1 mile = 1.609Kilometre
Relative Motion can be defined in the following ways :
Dependent as in the case of boats and streams.
Independent as in the case of trains
When one body which is moving inside another moving body, the motion is known as dependent, e.g When a boat is moving inside a stream, the speed of boat depends on the speed of stream. If boat is going in the same direction as the stream, then the boat will move faster than the speed at which boat is moving in still water, but when the boat is moving in opposite direction, the speed of boat is slower than speed of boat in still water.
BOATS AND STREAMS We consider four different types of speed :
Sd = speed downstream or speed of boat with the stream
Sup = speed upstream or speed of boat against the stream
Sb = speed of boat in still water.
Ss = speed of stream or water or river or current.
Sd = Sb + Ss...........(i)
Sup = Sb - Ss..........(ii)
Adding equation (i) and (ii) we get 2Sb = Sd + Sup
Sb = (Sd + Sup)/2 ......(iii) If we substract equations (ii) from equation (i), we get
2Ss = Sd - Sup
Ss = (Sd - Sup)/2.........(iv)
Example 1 A boat goes 24Km upstream and 36Km downstream taking 12 hours while it goes 36Km upstream and 24Km downstream in 13 hours.What is respective speed of boat and speed of stream?
Solution Let speed of boat be x and that of stream be y. Speed upstream = x - y
Speed downstream = x + y
Example 2 If Anshul rows 15Km upstream and 21 Km downstream taking 3 hours each time, then what is the speed of stream ?
QUESTIONS ON TRAINS
Let two trains or two bodies are moving in the same direction at u kmph and v kmph such that(u>v) then their relative speed is (u-v) kmph.
Continued...
Let two trains or two bodies are moving in the opposite direction at u kmph and v kmph such that(u>v) then their relative speed is (u+v) kmph.
If two trains of length x metres and y metres are moving in same directions at u kmph and v kmph then time taken by faster train to cross slower train = (x + y)/(u-v)
If two trains of length x metres and y metres are moving in opposite directions at u kmph and v kmph then time taken by faster train to cross slower train = (x + y)/(u+v)
Example 1:- A train 150m long is running at a speed of 50Km/h. How long will it take to cross a telegraph post?
Solution:- When train has to cross a stationary object, it has to cover a distance equal to its own length In this case the distance covered = 150m =3/20Km Speed of the train = 50Km/h 50Km is covered in = 60 minutes 1 Km is covered in = 60/50 minutes 3/20 Km is covered in = ( 60 * 3)/(50 * 20) or 9/20 minutes = (9 * 60)/50 or 10.8 seconds
Example 2:- The speed of a train, 125m long is 45Km/h. How much time will it take to pass a platform 625m long?
Solution When a train has to cross a stationery object which has same length it has to cover the sum of its own legth and the length of the stationary object.
In this case the distance covered = (125 +625)m = 750m
Speed 45Km/h = (45x1000)/60x60 m/sec = 25/2 m/sec
Time taken = (750x2)/25 sec = 60sec = 1minute
Example 3:- Two train 50m and 110m long are going at 34Km/h and 30Km/h respectively in opposite direction. How long would it take them to pass each other.?
Solution:- When two train go in opposite directions, we add their speed to get speed per hour. The direction covered is the sum of their length.
Now in this case
Speed per hour = 34 + 30 = 64Km = 64000m
Distance covered = 50 + 110 or 160m
Time taken to pass each other = 160/64000 or 1/400 hour
= 1*60*60/400 or 9seconds
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