algerbaic sys

Contents

1. INTRODUCTION
2. SOLVED EXAMPLES

Algebraic Symbols

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INTRODUCTION

In mathematics, the four basic operations, addition, subtraction, multiplication and division are known as fundamental operations and are represented by standard symbols +, , and respectively. But many times some special symbols are used to represent a mathematical operation.

For example, if we represent the average of two numbers a and b by (a $ b) then the result of (3 $ 6) is equal to (3 + 6)/2 or 4.5.

We will refer to these types of symbols as algebraic symbols.

To solve problems related to symbols we will use the concepts of binary operations (a calculation involving two operands). A binary operation, for example (a $ b), depends on the ordered pair (a, b). So to find ((a $ b) $ c), first operate on the ordered pair (a, b) and then operate on the result of that using the ordered pair ((a, b), c).

For example, if we represent the LCM of two numbers m and n by (m # n), then the result of

((2 # 3) # 8) will be ((6) # 8) = 24.

SOLVED EXAMPLES

Example 1:

If a $ b is the average of a and b, and m # n is the LCM of m and n, then find the value of (3 $ (4 # 5)). Solution: Solving the inner bracket, (4 # 5) = LCM of 4 and 5 = 20 ((3 $ (4 # 5)) = (3 $ (20)) (3 $ 20) = Average of 3 and 20 = (3 + 20)/2 = 23/2 = 11.5

Example 2:

If & (a, b) = the remainder when a is divided by b # (a, b) = number of ways to select a items from b distinct items $(a, b) = a if a > b or b if b > a then, find the value of $(#(&(7, 4), 5), 9). Solution: $(# (& (7, 4), 5), 9) = $(# (3, 5), 9) … [ & (7, 4) = 3] = $(10, 9) … [ # (3, 5) = 5C3 = 10] = 10 … [ $(10, 9) = 10]

Example 3:

A, B, C are three numbers. Let @ (A, B) = Average of A and B, / (A, B) = Product of A and B, and X (A, B) = The result of dividing A by B [CAT 2000] Question 1: The sum of A and B is given by (1) /(@( A, B), 2) (2) X(@(A, B), 2) (3) @(/(A, B), 2) (4) @(X(A, B), 2) Solution: Sum of A and B = (Average of A and B) 2 = /(@(A, B), 2) Hence, option 1. Question 2: Average of A, B and C is given by (1) @(/(@(/(B, A), 2), C), 3) (2) X(@(/(@(B, A), 3), C), 2) (3) /(@(X(@(B, A), 2), C), 3) (4) /(X(@(/(@(B, A), 2), C), 3), 2) Solution: We need to find average of A, B and C i.e. (A + B + C)/3 The only operator that can give 3 in the denominator is X. Option 1 cannot be the answer, as there is no X. Option 2 also cannot be the answer as X(@(/(@(/(B, A), 2), C), 3), 2) gives 2 in the denominator and not 3. Similarly option 3 is not the answer. Hence, option 4 must be the answer. Option 4: /( X (@ (/ (@ (B, A), 2), C), 3), 2) =/(X(@((A + B), C), 3), 2) Hence, option 4.

Example 4:

Let x and y denote two numbers. Then, $(x, y) = Average of x and y, &(x, y) = Product of x and y, and Q(x, y) = The result of dividing x by y If z denotes a third number, then the average of x, y and z is given by: (1) & ($(Q(&(x, y), 2), z), 3) (2) Q (& (Q(& ($(x, y), 3), z), 3), 2) (3) & (Q($(& ($(x, y), 2), z), 3), 2) (4) Q($(& ($(&(x, y), 2), z), 3), 2) Solution: The best way to solve this is by working through the options: Option 1: & ($(Q(&(x, y), 2), z), 3) &(x, y) = xy Q(xy, 2) = xy/2 Hence, option 1 is eliminated. Option 2: Q(&(Q(&($(x, y),3), z),3),2) (It should be apparent by now that this option will not lead to the required answer.) Hence, option 2 is also eliminated. Option 3: &(Q($(&($(x, y), 2), z), 3), 2) which is the average of x, y and z. Hence, option 3.

Example 5:

Two binary operations and * are defined over the set {a, e, f, g, h} as per the following tables a e f g h a a e f g h e e f g h a f f g h a e g g h a e f h h a e f g * a e f g h a a a a a a e a e f g h f a f h e g g a g e h f h a h g f e Thus, according to the first table f g = a, while according to the second table g * h = f, and so on. Also, let f2 = f * f, g3 = g * g * g, and so on. [CAT 2003 – Re-test] Question 1: What is the smallest positive integer n such that gn = e? (1) 4 (2) 5 (3) 2(4) 3 Solution: g2 = g * g = h g3 = g * g * g = h * g = f g4 = g * g * g * g = f * g = e Hence, option 1. Question 2: Upon simplification, f [f *{f (f * f)}] equals (1) e (2) f(3) g(4) h Solution: f * f = h f h = e f * e = f f f = h Hence, option 4. Question 3: Upon simplification, {a10 * (f10 g9)} e8 equals (1) e(2) f(3) g (4) h Solution: f10 = (f2)5 = h5 = h * (h2)2 = h * e2 = h * e = h g9 = g * (g2)4 = g * h4 = g * e = g Also, e8 = e ( e * e = e) and a10 = a ( a * a = a) Now, f10 g9 = h g = f a10 * f = a * f = a a e8 = a e = e Hence, option 1.

Example 6:

A florist hires some workers who can stack flowers in her store within 10 days. Which of the following gives the fraction of work the men do in a day? Symbols used: a @ b is the average of a and b. a & b is the product of a and b. a # b is the sum of the reciprocals of a and b. a $ b is the difference between the reciprocals of a and b. a ! b is the result when a is divided by b. (1) (((2 @ 3) & (2 ! (3 # 3))) $ ((2 & 2) & 2)) (2) ((((2 @ 3) ! (3 $ 2)) # 2) & (3 & 3)) (3) (((2 @ 3) ! (3 ! 2)) $ 2) (4) ((2 & (3 @ (2 # (2 ! (3 & 3)))))) Solution: If a certain number of men do some work in 10 days, then the same number of men will do 1/10th of the work per day. Going through each option, we find that only option 3 gives the answer as 1/10: (((2 @ 3) ! (3 ! 2)) $ 2) Hence, option 3.

REMEMBER:

• It is not necessary that the questions on algebraic symbols are always based on binary operations (or operations on two operands). There may be questions with operations on three or more operands.

For example, if we represent the geometric mean of three numbers a, b and c by @ (a, b, c), then,

@ (2, 4, 27) = (2 4 27)^1/3 = (8^1/3 27^1/3) or (2 3) = 6

Example 7:

Aaron is Babitti’s husband. Together, they have two daughters; Corinne and Deborah. Deborah is married to Xerxis and they have a beautiful baby girl named Ginger. It so happens that Xerxis’s parents, Yates (father) and Zara (mother), are very close friends of Aaron and Babitti. Also, their (i.e. Yates and Zara’s) daughter, Winnie, is married to Pierre and they have a son named Eomer. Notations: a $ b: a is b’s sister a # b: a is b’s brother a & b: a is b’s wife a b: a is a parent of b a ← b: a is a child of b a ^ b: a is b’s sister-in-law a % b: a is b’s brother-in-law Using the above notations, for example, a $ b & c would mean that a is b’s sister, who in turn is c’s wife. Which of the following statements cannot be derived from the above information? (1) Corinne $ Deborah & Xerxis ← Zara & Yates Winnie Eomer (2) Ginger ← Xerxis % Corinne $ Deborah ^ Winnie & Pierre Eomer (3) Zara & Yates Xerxis # Winnie % Deborah $ Corinne ← Babitti & Aaron (4) All the above statements are true and can be derived. Solution: We will use only the first letter of each person to denote them. (M) and (F) denote Male and Female respectively. According to the information given: Now, let’s work through the options: Option 1: C is D’s sister; D is X’s wife; X is a child of Z; Z is Y’s wife; Y is a parent of W; W is a parent of E. All of these are from the information given in the passage; hence, option 1 can be derived. Option 2: G is a child of X; X is C’s brother-in-law; C is D’s sister; D is W’s sister-in-law; W is P’s wife; P is a parent of E. All of these are from the information given in the passage; hence, option 2 can be derived. Option 3: Z is Y’s wife; Y is the parent of X; X is W’s brother; W is D’s brother-in-law. Winnie is Deborah’s sister-in-law, not brother-in-law. This statement is false. Hence, option 3.