1. Amity School of Business 1 Amity School of Business BBA Semester IV ANALYTICAL SKILL BUILDING

2. Amity School of Business 2 Ratio Ratio is simply the quotient of two numbers. This is where we get the word "rational numbers." A rational number is any number that can be expressed as the ratio or quotient of two integers (denominators cannot equal zero). Every time you write a fraction, you have written a ratio. A proportion is simply the equating of two ratios. Whenever one ratio (or fraction) equals another ratio (or fraction), this is a proportion

3. Amity School of Business 3 Ratio To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions. Example: Are the ratios 3 to 4 and 6:8 equal? The ratios are equal if 3/4 = 6/8. These are equal if their cross products are equal; that is, if 3 x 8 = 4 x 6. Since both of these products equal 24, the answer is yes, the ratios are equal.

4. Amity School of Business 4 Ratio A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon ":".

5. Amity School of Business 5 Ratio A comparison of numbers with the same units so units are not required. 3:9 4/12 5 to 20 Ratio1:3 <1/3

6. Amity School of Business 6 Rate A comparison of 2 measurements with different units. Example Unit 10 km per 2 h $ 6 per 3 h Rate 5 km per h <$ 2/h

7. Amity School of Business 7 Few facts Duplicate ratio of a : b a 2 : b 2 Triplicate ratio of a : ba 3 : b 3 Subduplicate ratio√a : √b Subtriplicate ratio 3 √a : 3 √b

8. Amity School of Business 8 Properties of ratio Invertendo Alternendo componendo Dividendo Componendo - Dividendo Equal ratio

9. Amity School of Business 9 Let x : y be a ratio, which can also be written as x/y. We will try to find out what will happen when a constant a is added both to the numerator and denominator CASE I (x/y < 1 ) If x/y<1, then addition of a constant positive number to numerator and denominator leads to a bigger ratio than the ratio itself, i.e. x/y < (a+x)/(a+y) for x/y<1 where a is a constant positive number. e.g. 1/2 is less than 1 and when we add 2 to both numerator and denominator we get 3/4 and 3/4 is greater than 1/2 Ratio

10. Amity School of Business 10 Ratio CASE I (x/y < 1 ) If x/y<1, then addition of a constant positive number to numerator and denominator leads to a bigger ratio than the ratio itself, i.e. x/y (x-a)/(y-a) for x/y<1 Lets consider a fraction 5/10, if 5 is subtracted from numerator as well as denominator, we get 0 and it is less than 5/10(i.e. 1/2) Thus the rule for the case of subtraction is the reverse of the case of addition, as can be easily seen by the given example

11. Amity School of Business 11 Ratio CASE II (x/y > 1 ) The above rule gets totally and exactly reversed for x/y >1. Therefore, x/y > (x+a)/(y+b) and x/y < (x-a)/(y-b).

12. Amity School of Business 12 Four (non-zero) quantities of the same kind a, b, c, d are in proportion, written as a : b :: c : d if a/b = c/d The non-zero quantities of the same kind a, b, c, d,... are in continued proportion if a/b = b/c = c/d =... In particular, a, b, c are in continued proportion if 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. Proportion

13. Amity School of Business 13 Proportion The four parts of the proportion are separated into two groups, the means and the extremes, based on their arrangement in the proportion. Reading from left-to-right and top-to-bottom, the extremes are the very first number, and the very last number. This can be remembered because they are at the extreme beginning and the extreme end. Reading from left-to-right and top- to-bottom, the means are the second and third numbers. Remembering that "mean" is a type of average may help you remember that the means of a proportion are "in the middle" when reading left-to-right, top-to-bottom. Both the means and the extremes are illustrated below.

14. Amity School of Business 14 Proportion Algebra properties tell us that the products of the means is equal to the product of the extremes fraction one-half is equal to two-fourths. This is shown as a proportion below. 1/2 = 2/4 2 * 2 = 1 * 4 4 = 4

15. Amity School of Business 15 Proportion A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. 3/4 = 6/8 is an example of a proportion. When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Question marks or letters are frequently used in place of the unknown number. Example: Solve for n: 1/2 = n/4. Using cross products we see that 2 x n = 1 x 4 =4, so 2n = 4. Dividing both sides by 2, n = 4 / 2 so that n = 2.

16. Amity School of Business 16 Proportion Used to compare two ratios or make equivalent fractions. To solve one can: 1.use equivalent fractions. 2.cross multiply. 1/2 = 3/4 1 x 4 = 2 x 3 Note: The product of the outside (first and last) numbers equals the product of the two middle numbers.

17. Amity School of Business 17 Mixtures Simple mixture – When two different ingredients are mixed together, it is known as simple mixture. Compound Mixture – When two mixtures are mixed together to form another mixture, it is known as compound mixture

18. Amity School of Business 18 Alligation When different quantities of the same or different ingredients of different costs (one cheap and other dear) are mixed together to produce a mixture of a mean cost, the ratio of their quantities is inversely proportional to the difference in their cost from the mean cost.

19. Amity School of Business 19 Alligation If a vessel contains ‘a’ litres of liquid A, and if ‘b’ liters be withdrawn and replaced by liquid B, then if ‘b’ litres of mixture be withdrawn and replaced by liquid B, and the operation repeated ‘n’ times in all. Then : Liquid A left after nth operation Liquid B left after nth operation

20. Amity School of Business 20 When X1 quantity of A of cost C1 and X2 quantity of B of cost C2 are mixed, then the cost of the mixture is When two mixtures M1 and M2, each containing ingredient A and B in the ratio a:b and x:y are mixed in the roportion of the ingredients A and B i.e. Qa:Qb in the mixture is

21. Amity School of Business 21 Alligation

22. Amity School of Business 22 A milkman with draws 1lit of milk from a vessel containing 10 lit of pure milk and replaces it with water. Lets see how the concentration changes after each operation. Initial amount of milk = 10 lit After first replacement, amount of milk = 9 lit and water = 1lit As per the formula, Amount of milk left / Initial amount of milk = 1 – 1/ 10 = 9/10 (same as above) Second operation: Now the vessel contains 9lt milk and 1lt water The1 lt mixture that ‘ll taken out of the container in second operation ‘ll contain 0.1 lt of water and 0.9lt of milk and that ‘ll be replaced by 1 lt of water So amount of milk in the mixture after second operation = 9 – 0.9 = 8.1 and amount of water in the mixture after second operation = 1 – 0.1 + 1 = 1.9 (=10- 81.) Now as per formula Amount of milk left / Initial amount of milk = (1 – 1/10)^2 = 81/100 (same as above) Notes: 1. Some time the amount of liquid taken out and replaced by are not same. In that case don’t use the formula. Calculate the final composition by the method explained above Problem

23. Amity School of Business 23 The intersection of a set and its complement is the empty set. For example, let A consist of all the unbroken plates in a set of plates, and let A' consist of all the broken plates. Then (A. A‘) is empty because there are no plates that are unbroken and broken at the same time. We summarize this law in symbols as follows: A. A' = 0. Intersection of a Set and its Complement

24. Amity School of Business 24 Venn diagrams Venn diagrams are illustrations used in the branch of mathematics known as set theory. They are used to show the mathematical or logical relationship between different groups of things (sets). A Venn diagram shows all the logical relations between the sets

25. Amity School of Business 25 Venn diagrams

26. Amity School of Business 26 Since the intersection of two sets A and B is given by A. B = {x: x is in A and x is in B}, the complement of this intersection is given by (A. B)' = {x: not-(x is in A and x is in B)}. But not-(x is in A and x is in B) has the same meaning as x is not in A or x is not in B, and {x: x is not in A or x is not in B} = A' + B'. Complement of an Intersection

27. Amity School of Business 27 we have the following law for the complement of an intersection: (A n B)' = A' u B'. This is another of De Morgan's laws for sets. For example, let the universal set be the whole numbers from 1 to 10. Let A = {1, 2, 3, 4, 7, 8, 9, 10} and let B = {2, 4, 6, 8, 10}. Then A. B = {2, 4, 8, 10}, and so (A. B)' = {1, 3, 5, 6, 7, 9}. Also A' = {5, 6}, B' = {1, 3, 5, 7, 9} and A' + B' = {1, 3, 5, 6, 7, 9}. Therefore, in this example, (A. B)' = A' + B'. Complement of an Intersection

28. Amity School of Business 28 Since the union of two sets A and B is given by A + B = {x: x is in A or x is in B}, the complement of the union is given by (A + B)' = {x: not-(x is in A or x is in B)} But not-(x is in A or x is in B) has the same meaning as x is not in A and x is not in B, and {x: x is not in A and x is not in B} = A'.B'. Complement of an Union

29. Amity School of Business 29 We have the following law for the complement of a union: (A + B)' = A'. B'. This is one of De Morgan's laws for sets. For example, let the universal set be the set of all substances. Let A be the set of all solids, such as stone and iron. Let B be the set of all liquids, such as water and oil. Then A + B is the set all substances that are solid or liquid. The complement (A + B)' is the set of all substances that are not solid or liquid, in other words the set A'. B' of all substances that are not solid and not liquid, such as oxygen and nitrogen (which are gases). Complement of an Union

30. Amity School of Business 30 The complement of a complement is written in symbols as follows: (A')' = {x: not-(x is not in A)}. Since not-(x is not in A) has the same meaning as x is in A, it follows that (A')' = {x: x is in A}. But {x: x is in A} = A. Therefore we have the complement law: (A')' = A. For example, let the universal set be the set of whole numbers from 1 to 10, and let A be the set of even numbers from 2 to 10. Then A' is the set of odd numbers from 1 to 9, and (A')' is the original set A of even numbers from 2 to Complement of a complement

31. Amity School of Business 31 The union of a set and its complement is the universal set. For example, let x stand for an animal, let A = {x: x is male}, and let A' = {x: x is female}. Then A + A' = {x: x is male or x is female} = the set of all animals. We write this law briefly as follows: A u A' = U. Union of a set and its complement

32. Amity School of Business 32 Distance between two points P(x1,y1) and Q(x2,y2) Coordinates of a mid point The coordinate of the point R(x,y) which divides a straight line joining two points (x1,y1) and (x2,y2) internally in a given ratio m1:m2 Coordinate geometry

33. Amity School of Business 33 Area of triangle whose vertices are (x1,y1), (x2,y2) and (x3,y3) is Coordinate geometry