The symbol ∆ denotes one of the four arithmetic operations:addition,su

Question

The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

generis · Accepted Answer

If 6 ∆ 3 ≤ 3, then the ∆ operator is subtraction AND division.

6 - 3 = 3, which satisfies the equation

\frac{6}{3}  = 2, which also satisfies the equation (2 is less than 3).

The operator cannot be addition or multiplication. 6 + 3 = 9, which is > 3. And 6 * 3 = 18, also > than 3.

Check I, II, and III.

I. 2 ∆ 2 = 0. If ∆ were only subtraction, this would be fine. 2 - 2 = 0.

But  \frac{2}{2}  = 1. This equation is not true when ∆ is division. Incorrect.

II. 2 ∆ 2 = 1. If ∆ were only division, this would be fine. \frac{2}{2}  = 1. But when ∆ is subtraction, then 2 - 2 = 0. This equation does not hold true when ∆ is subtraction. Incorrect.

III. 4 ∆ 2 = 2. This must be true for both subtraction and division.

It holds for subtraction. 4 - 4 = 2 
And it holds for division:  \frac{4}{2} = 2. Correct.

Answer C

BrentGMATPrepNow · Answer

The above solutions are perfect. 
I just want to point out that, when answering this type of question type (where the answer choices are I only, II and III only, etc.), be sure to  ELIMINATE answer choices after examining each statement.

So, for example, once we determine that statement I need not be true, we can ELIMINATE answer choices A, D and E
Then, once we determine that statement II need not be true, we can ELIMINATE answer choice B

At this point, there's no need to check statement III, since we already know that the correct answer must be C

Cheers,
Brent

sashiim20 · Answer

6∆3≤ 3

\frac{6}{3}= 3

6-3 = 3

Therefore symbol ∆ should be division and subtraction both.

Checking the options, we get;

I. 2∆2=0

2 - 2 = 0

\frac{2}{2}= 1 ------------ Must Not be true.

II . 2∆2=1

\frac{2}{2}= 1

2 - 2 = 0  --------------- Must Not be true.

III . 4∆2=2

\frac{4}{2} = 2  --------- Must be true.

4 - 2 = 2  --------- Must be true.

Hence Only III must be true. Answer (C)...

elegantm · Answer

∆ could be either Subtraction or Division in order to fulfill condition 6∆3≤ 3
I - 2∆2=0 - In this condition, ∆ denotes only subtraction and not Division. Hence Wrong
II - 2∆2=1 - In this condition, ∆ denotes only Division and not subtraction. Hence Wrong
III - 4∆2=1 - In this condition, ∆ denotes either Division or subtraction. Hence Correct.

hence, Answer C

RaguramanS · Answer

From the context,  ∆= subtraction and division 
∆= subtraction 6-3<=3  
∆= division \frac{6}{3}=2<=3

From the options
Only iii holds true as 4∆2=2( both division and subtraction)

Option c

JeffTargetTestPrep · Answer

Let’s analyze the given inequality:

If ∆ = addition:

6 + 3 ≤ 3

9 ≤ 3

Since 9 is not less than 3, ∆ cannot be addition.

If ∆ = subtraction:

6 - 3 ≤ 3

3 ≤ 3

Since 3 is less than or equal to 3, ∆ can be subtraction.

If ∆ = divison:

6/3 ≤ 3

2 ≤ 3

Since 2 is less than or equal to 3, ∆ can be division.

If ∆ = multiplication:

6 x 3 ≤ 3

18 ≤ 3

Since 18 is not less than or equal to 3, ∆ cannot be multiplication.

We see then that ∆ is either subtraction or division. Now let’s analyze each Roman numeral:

I. 2∆2=0

Since 2/2 does not equal 0, division doesn’t work and I is not true.

II. 2∆2=1

Since 2 - 2 does not equal 1, subtraction doesn’t work and II is not true.

III. 4∆2=2

Since 4 - 2 = 2 and 4/2 = 2, III is true.

Answer: C

shashankism · Answer

6∆3≤ 3

So, 6 - 3 = 3
6/3 = 2

I. 2∆2 
2-2 = 0 
2/2 = 1
So it may be true.

II. 2∆2
2-2 = 0 
2/2 = 1
So it is may be true.

III. 4∆2
4-2 = 2
4/2 = 2 
So, 4∆2 = 2 must be true.

Answer C.

EMPOWERgmatRichC · Answer

Hi All,

We're told that the symbol # denotes one of the four arithmetic operations: addition, subtraction, multiplication, or division and that 6#3 <= 3. We're asked which of the three Roman Numerals must be true.

To start, let's do a bit of analysis on the inequality that we're given (and determine which math operation COULD fit this): 
6#3 <= 3

6 + 3 = 9 which is NOT <= 3 
6 - 3 = 3 which IS <= 3 
(6)(3) = 18 which is NOT <= 3 
6/3 = 2 which IS <=3

Thus, the symbol # must be either subtraction or division (although we don't know which one for sure).

I. 2#2 = 0

We've proven that the symbol is either subtraction or division, so let's see if either of those operations 'fits' this information... 
2 - 2 = 0 
2/2 = 1 
Thus, if the symbol is subtraction, then Roman Numeral 1 IS true. However, if the symbol is division, then Roman Numeral 1 is NOT true. There's no way to know which symbol is involved though, so Roman Numeral 1 is not necessarily true. 
Eliminate Answers A, D and E.

II. 2#2 = 1

With the work that we've done in Roman Numeral 1 (above), we already know that the outcome of 2#2 could be 0 or 1. This inconsistency also provides that Roman Numeral 2 is not necessarily true. 
Eliminate Answer B.

There's only one answer remaining....

III. 4#2 = 2

You can still prove that Roman Numeral 3 is always true... 
4 - 2 = 2 
4/2 = 2 
Both outcomes 'fit' the information in Fact 3, so regardless of what operation is represented, the statement IS true.

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

RJ7X0DefiningMyX · Answer

Why does the question say "denotes  one of the  four arithmetic operations", wherein while solving, ∆ actually denotes  two  operations (both subtraction and division).

Coming from the makers of the GMAT itself, isn't the language a bit unclear?

EMPOWERgmatRichC · Answer

Hi rishabhjain13,

Most GMAT questions (in both the Quant and Verbal sections) require 3-5 'steps' to solve - and in many cases, at least one of the steps requires that you properly 'link' one piece of information to another and make a deduction. As a simple example, you might see a question that states A and B are integers and B > 0. The deduction here is that A can be any type of integer (negative, 0 or positive), while B can only be a positive integer. You might be thinking "why not just say that B is a positive integer?".... but that removes one of the essential elements of the question (and of the overall Exam): measuring how well you can link ideas and make deductions.

The same concept applies to this question. We start off with a broader piece of information (re: there are 4 possible math functions, but which one is it?) and then we're given a second piece of information that narrows down the possibilities (re: it's either subtraction or division, but we still don't know exact which one until we move on to the next part of the question).

GMAT assassins aren't born, they're made,
Rich

jabhatta2 · Answer

VeritasKarishma  |  Bunuel

As a test taker -- how are we supposed to know that ∆ should represent BOTH division AND subtraction

I assumed, the conditions applied individually (subtraction  OR    division) and thus opted for E

It's clear  one or the other works

Since the question says 6∆3=<3 
6/3 = 2
OR 
6-3 = 3

--------------

KarishmaB · Answer

∆ will represent one operation only - either division or subtraction. Which one it actually represents, we do not know. So whichever statement MUST BE TRUE should be the one in which ∆ could represent either operation, it would hold.

I. 2∆2 = 0
Is true if ∆ is subtraction but not if it is division.

II. 2∆2 = 1
Is true if ∆ is division but not if it is subtraction.

III. 4∆2 = 2
4 - 2 = 2
4/2 = 2
So this will hold irrespective of whether ∆ is subtraction or division. So this statement will definitely be true.

sam12rawat · Answer

Hello, I have a query regarding the answer. The question states that the the symbol denoted in the question stem must be any ONE of the four operations and though I too agree that given the example the answer choice should hodl true in subtraction AND division, I dont understand how it could be two operations even though its mentioned it should be any ONE. Thank you.

EMPOWERgmatRichC · Answer

Hi sam12rawat,

When a Quant question asks "which of the following MUST be true?", what it's really asking is "which of the following is ALWAYS true no matter how many different examples we can come up with?"

In the case of this Roman Numeral question, we have to consider whether each of those 3 equations is ALWAYS true or not. Even though the 'symbol' might be subtraction or division (we can never actually define which one), only Roman Numeral 3 is ALWAYS true (whether the symbol is subtraction or division).
 
GMAT assassins aren't born, they're made,
Rich

CrackverbalGMAT · Answer

SOLUTION:

This is a question from Function and custom characters.

As the operation does not increase the value to 9 or 18,it is possibly an operation of division(/) OR multiplication(*).

Now,6-3 = 3 and 6/3=2 (which is less than 3)

=>Both the operations are possible to represent the operation.

Let us check the options now:

I . 2∆2 = 0 = > 2-2 = 0 yields the result but 2/2 doesn't and hence it is NOT True always

II. 2∆2 = 1=>2/2 = 1 yields the result but 2-2 doesn't and hence it is NOT True always
(ELIMINATE A,B,D,E at this stage and arrive at C as ANSWER)

Also, III. 4∆2 = 2=>4-2 = 2 and 4/2=2 yields the same result and hence it is ALWAYS True OPTION(C)

Hope this helps

Devmitra Sen(Math)

GMATinsight · Answer

Answer: Option C

Video solution by  GMATinsight

https://www.youtube.com/watch?v=XQPGIeDAj6E

mayankparihar1 · Answer

Why is E wrong?

The equation suggests 6∆3≤ 3 ie., 6∆3 is either less than 3 or equal to 3

so, why wouldn't choosing option E solve all of the conditions? Don't we have to just pass either of the two conditions?

EMPOWERgmatRichC · Answer

Hi mayankparihar1,

When a Quant question asks "which of the following MUST be true?", what it's really asking is "which of the following is ALWAYS true no matter how many different examples we can come up with?"

In the case of this Roman Numeral question, we have to consider whether each of those 3 equations is ALWAYS true or not. Even though the 'symbol' might be subtraction or division (although we are not certain for sure which one), only Roman Numeral 3 is ALWAYS true (whether the symbol is subtraction or division).

GMAT assassins aren't born, they're made,
Rich

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=x0flhI0YIP0

The symbol ∆ denotes one of the four arithmetic operations:addition,su

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