Ratio, Proportion, Variation Aptitude basics, practice questions, answers and explanations 
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Ratio Proportion and Variation

Ratio: The ratio a : b represents a fraction a/b. a is called antecedent and b is called consequent. Ratio is the relation between two numbers which is expressed by a fraction.The equality of two different ratios is called proportion.
  
If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.

In a : b = c : d, then we have a* d = b * c.

If a/b = c/d then ( a + b ) / ( a - b ) = ( d + c ) / ( d - c ). The ratio of two quantities a and b in the same units, is the fraction  a/b and we write it as a : b. Ratio of two quantities is always an abstract number (without any units).

In the ratio a : b, we call 'a' as the first term or antecedent and 'b', the second term or consequent.

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.
 

Tips: If A is thrice as good a workman as B, then Ratio of work done by A and B = 3 : 1.
Ratio of times taken by A and B to finish a work = 1 : 3

General formula can be extended if more than 2 people (or machines are working together) 1/TA+1/TB+1/TC+......= I/Ttogether; where TA,TB and TC are the times taken by A, B and C respectively to complete the task alone and Ttogether is the time taken by them to complete the task when they are all working together.

Eg: If Alex can build a house in 2 days and his apprentice Bob can build a house in 3 days, then how long will it take  Alex and Bob to build a house when they are working together?

Putting the information from the question into the formula gives us,
Invert both sides of the equation Time working together=6/5=11/5 days. So Alex and Bob will take 11/5 days to build a house when they are working together.
 

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. If a : b : : c : d, then a + b : b : : c + d : d
    2. If a : b : : c : d, then a – b : b : : c – d : d
    3. If a : b : : c : d, then a + b : a – b : : c + d : c – d
    4. If a/b = c/d = e/f = …..k, then k = (a ± c ± e ……)/ b ± d ± f ……

Tips: 1)  Direct proportion: If x is directly proportional to y:  x1/y1 = x2/y2
         2)  Indirect proportion: If x is inversely proportional to y:  x1y1 = x2y2
 

Variation:
We say that x is directly proportional to y, if x = ky for some constant k and we write,
x   1/y

Chain Rule

Direct Proportion: Two quantities are said to be proportional, if on the increase (or decrease) of the one, the other increases ( or decreases) to the same extent.

Eg. 1) Cost is directly proportional to the number of articles
(More articles, More Cost)
Eg. 2) Work done is directly proportional to the number of men work (More Men, More Work)

Indirect proportion : Two quantities are said to be indirectly proportional, if on the increase of the one, the other decreases to the same extent and vice-versa.

Eg. 1) The time taken by a car is covering a certain distance is inversely proportional to the speed of the car. (More speed, Less is the time taken to cover a distance)

Eg. 2) The time taken to finish a work is inversely proportional to the number of persons working at it. (More persons, Less is the time taken to finish a job)

Remarks: In solving problems by chain rule, we compare every item with the term to be found out.

Key Notes:

If p:q : : r:s
=> s= qr/p
The method of finding the 4th term of a proportion when three are given is known as rule of three as above.
Three or more quantities are said to be in compound proportion if one quantity depends on the other remaining quantities.

If p,q,r,s are four quantities & if p:q : : r:s, then

1) Componendo
(p+q)/ q = (r+s)/s
 

2) Dividendo
(p-q)/p = (r-s)/s

3) Componendo & Dividendo
(p+q)/ (p-q)   = (r+s)/ (r-s)
 

4) Invertendo
q/p = s/r

5) Alternendo
p/r = q/s

Direct proportion is indicated by arrows in the same direction, Inverse proportion is indicated by arrows in opposite direction.