Bill understood the meaning of the engravings. He had a better idea of the nature of the occupants of “The House of Rational Numbers, Q”. He saw another stone which had the following message engraved:

"This hall would always be associative for any occupants of the "House of Rational Numbers, Q""

This stone could correspond to which of the four halls?

Addition

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Subtraction

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Multiplication

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Division

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Solution

The correct options are
A
Addition

C
Multiplication

Rational numbers are associative for addition and multiplication as (A +B) +C = A +(B +C) and also, $(A \times B) \times C = A \times (B \times C)$
Rational numbers are not associative for subtraction and division as (A - B) - C is not be equal to A - (B - C) and also, $(A \div B) \div C$ is not be equal to $A \div (B \div C)$

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Similar questions

Bill thought of giving “The Hall of Operations” a visit, after listening to these interesting behaviors of the halls towards occupants from different rooms and houses.

He happened to visit the Hall from the back door entry and he saw 4 huge stones, one against each of the 4 halls (Addition, Subtraction, Multiplication and Division). Each stone had engravings on it describing the hall against which it is erected.

On one such stone, the following was engraved

“This hall has been and forever will be commutative for EVERY occupant of the “House of Rational Numbers, Q””.

This stone could correspond to which of the four halls?

Bill was curious to find the room into which the previous number went in (which exited the “Hall of Division” when occupants “7” and “8” went in). He came out of the room he was in and was surprised at what he saw!

He noticed that the previous rooms he saw – Room N, Room W and Room I were all inside a huge house whose nameplate read – “House of Rational Numbers, Q”. This house of rational numbers was part of a huge villa – many times larger. Bill couldn’t see the nameplate of this villa.

He realized that the arrangements of the Rooms N, W and I inside “House of Rational Numbers, Q” were in logical accordance with the nature of numbers in Earth 1. What could Bill’s inferences be? Choose ALL the correct inferences Bill should have made.

Bill wanted to find out the room into which the exited number went in. When he zoomed into the same premises of “The Hall of Operations” using his binoculars, he saw the following engravings on the walls of the halls:

Engraving 1: All natural numbers are rational numbers

Engraving 2: All natural numbers are rational numbers

Engraving 3: All integers are rational numbers

Help Bill by validating the engravings by selecting ALL the correct engravings, given in the options below:

Bill heard that in “The Hall of Operations”, there was another ancient temple, named “The Reciprocus”.

Legend says that when an occupant, say “a”, of the “House of Rational Numbers, Q” enters the temple, it exits the temple as a different occupant, “b”,belonging to the same house. Bill observed that, these occupants “a” and “b”, when they enter the hall of multiplication, they exit the hall as “1”.

Bill realized that the numerical counterparts of a and b (Earth 1) are ___________ of each other.

From inside the Room I, Bill noticed a huge commercial space, “The Hall of Operations”, which contained four halls named Addition, Subtraction, Multiplication and Division. These halls were named according to the operation being carried out. Each of these halls had 2 entries and only 1 exit.

Bill observed that when any two occupants (numbers) entered the room through the entries, the operation specified by the hall happened on the numbers in the order they enter and from the exit, came the result.

Bill noticed occupants from Room I enter “The Hall of Operations” in pairs.

From which of the halls would the occupant coming out from the exit be from Room I? [Select ALL the relevant options]

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