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The symbols @ and # each denote one of the four arithmetic operations: addition, subtraction, multiplication and division. If the symbols denote different arithmetic operations such thatand a@b#c = c#b@a, where a, b and c are distinct positive integers, what is the value of 4@5#4?
- a)-24
- b)-16
- c)16
- d)24
- e)Can't be determined
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The symbols @ and # each denote one of the four arithmetic operations:...
Given
- @ = { + , - , *, /}
- # = { + , - , *, /}
- @ ≠ #
- a@b#c = c#b@a, where a, b , c are distinct integers > 0
To Find: value of 4@5#4?
- We need to find the operations that @ and # denote.
Approach
- To find the operations that @ and # denote, we will analyse the relation a@b#c = c#b@a.
- We will calculate the values of 4@5#4 for all the possible operations denoted by @ and #.
Working Out
- As a@(b#c) = (c#b)@a, we can say that @ = {*, +}
- Similarly, as (a@b) # c = c#(b@a), we can say that # = {+, *}
- Now, let’s evaluate all the possible cases for the operations denoted by @ and #.
- {@, #} = {*, +} or {*, *} or {+, *} or {+, +}.
- Since @ and # denote different operations, {*,*} and {+, +} are not the possible cases.
- Evaluating the possible cases, we have:
- If @ = * and # = +
- So, 4@5#4 = 4*5 + 4 = 24
- If @ = + and # = *
- So, 4@5#4 = 4+5*4 = 24
- If @ = * and # = +
Thus, we have a unique value of 4@5#4 = 24
Answer: D
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