Ratio-Proportion-Variation Notes for Competitive Exams Preparation
Ratio:
A ratio a : b can also be expressed as a/b. So if two items are in the ratio 2 : 3, we can say that their ratio is 2/3. If two terms are in the ratio 2, it means that they are in the ratio of 2/1, ie., 2 :1.
" A ratio is said to be a ratio of greater inequality of less inequality or of equality according as antecedent is greater than, less than or equal to consequent". From this we find that a ratio of greater inequality is diminished and ratio of less inequality is increased by adding same quantity to both terms.
ie., in a : b
If a<b then (a+x) : (b+x)> a: b and a>b then (a+x):(b+x) < a:b
if a/b=c/d=e/f ...., then each of these ratio is equal to a+c+e+..../b+d+f+...
Proportion:
When two ratios are equal the four quantities involved in the two ration are said to be proportional
ie., If a/b= c/d, then a, b, c and d are proportional. This is represented as a:b :: c:d and si read as "a is to b (is) as c is to d".
When a, b, c and d are in proportion, then the items a and d are called the EXTREMES and the items b and c are called the MEANS. We also have the relationship.
Product of the MEANS = Product of the EXTREMES ie., bc= ad
If a:b = c:d then
b:a = d:c----(1)
a: c= b:d
a+b: b= c+d:d....(2) obtained by adding 1 to both sides of the given relationship
a-b:b=c-d:d.....(3) obtained by subtracting 1 from both sides of the given relationship
a+b:a-b=c+d:c-d .....(4) obtained by dividing relationship (2) above by (3).
Relationship
(1) above is called INVERTENDO
(2) is called COMPONENDO
(3) is called DIVIDENDO and
(4) is called COMPONENDO- DIVIDENDO.
The last relationship, ie., COMPONENDO-DIVIDENDO is very helpful in simplifying problems.
Whenever we know a/b=c/d, then we can write (a+b)/(a-b)=(c+d)/(c-d) by this rule. The converse of this is also true whenever we know that (a+b)/(a-b)= (c+d)/(c-d), then we can conclude that a/b=c/d.
If three quantities a, b and c are such that a:b :: b:c , then we say that they are in CONTINUED PROPORTION. We also get b square= ac
Variation:
Two quantities A and B may be such that as one quantities changes in value, the other quantity also changes in value depending on the change in the value of the first quantity.
Direct Variation:
One quantity A is said to vary directly as another quantity B if the two quantities depend upon each other in such a manner that if B is increased in a certain ration, A is increased in the same ratio and if B is decreased, A is decreased in the same ratio.
This is denoted as A alpha B then A= kB, where k is a constant. It is called a constant of proportionality.
Inverse Variation:
A quantity A is said to vary inversely as another quantity B if the two quantities depend upon each other in such a manner that if B is increased in a certain ratio, A is decreased in the same ratio and if B is decreased then A is increased in the same ratio. It is the same as saying that A varies directly with 1/B. It is denoted as if A alpha 1/B i.e., A=k/B, where k is a constant of proportionality.
Joint Variation:
If there are three quantities A, B, and C such that A varies with B when C is constant and varies with C when B is constant, then A is said to vary jointly with B and C when both B and C are varying.
then A alpha BC or A=kBC, where k is the constant of proportionality.
Worked Out Examples:
1) Two numbers are in the ratio 9: 11. If 5 be added to each, then the numbers will be in the ratio 5: 6. Find the numbers.Sol:- x/y= 9/11
Let x= 9k, y= 11k
9k+5/11 k+5=5/6
54k+30=55 k+25
k= 5
x=9(5)=45
y=11(5)= 55
2) The volume of cylinder varies jointly as its height and the area of its base. When the area of the base is 300 square feet and the height is 6 feet the volume is 600 cu.ft. what is the area of the base of a cylinder whose volume is 360 cu.ft. and height is 12 ft.
Sol:- Let v be the volume, a be the area of base, h be the height.
V=m a h, where m is the constant.
Therefore, 600=m(300)(6)
1/3=m
therefore V=1/3(a)(h)
Now, 360 = 1/3(a)(12)
a= 90 sq.ft.
Important Links:
- Computer Knowledge Notes for Bank Exams Preparation
- Marketing and Business Knowledge Notes for Bank Exams Preparation
- Probability Notes for Competitive Exams Preparation
0 comments: