If 2a – b = 3c, where a, b, and c are non-zero integers

Question

If 2a – b = 3c, where a, b, and c are non-zero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?

A. -2
B. -1 
C. 1
D. 10
E. 12

Bunuel · Accepted Answer

2a-b=3c  -->  b=2a-3c .

Now,  average(a,b)=\frac{a+b}{2}=\frac{a+(2a-3c)}{2}=3*\frac{a-c}{2}=integer , so average is a multiple of 3 (as given that average and all unknowns are integers). Only multiple of 3 among answers is 12.

Answer: E.

Hope it's clear.

Topaz · Answer

Hi Bunuel,

I have a different approach to this question, Please let me know if It is correct...

Lets say Average of a & b is c

a+b/2=c
a+b= 2c

2a-b=3c
 a+b=2c
--------
3a = 5c

a=5c/3

we can plug in all answer choices for average - " c " and see which one gives us an integer as "a" is also an integer.

5(-2)/3 = -10/3
5(-1)/3 = -5/3
5(1)/3 = 5/3
5(10)/3 = 50/3
5(12)/3 = 60/3= 20 ( Hence E- 12 is the correct answer)

prashantbacchewar · Answer

I also got E by using substitution approach. However Bunuel suggested a good approach. Thanks

tingle15 · Answer

My solution:

2a - b = 3c
2a + 2b - 2b - b = 3c
2(a + b) = 3(b + c)
(a + b)/2 = average = (3/4) (b + c)

For average to be an integer, it must be a multiple of 12. Hence E

Bunuel · Answer

Merging similar topics.

For the question above there are two correct answer A (for example a=9, b=15, c=1) and E (for example a=3, b=3, c=1), as both are multiple of 3.

amit2k9 · Answer

2a-b/3 = c integer.

a+b = 2 * average

average = 12 
a+b = 24

3(a-8)/3 = integer.

hence E

IEsailor · Answer

E through substitution though its time consuming. Bunuel has the best approach

BangOn · Answer

checking options....

a+b = 24/20/2/-2/-4

-b = a - 24/20/2/-2/-4

Now, 2a - b = 3 c

only 24 gives us a multiple of 3 that can be taken out.

so E.

KarishmaB · Answer

Responding to a pm:

Bunuel has already given the algebraic approach which is quite simple and clear. I am guessing that since you are looking for another approach, you want me to solve it without using algebra.

I instinctively jumped to the options in this question. They are the average of a and b so sum of a and b will be twice of the average so depending on which option we pick, the sum (a + b) will be -4 or -2 or 2 or 20 or 24. Note that a and b will be either both even or both odd since their sum must be even.

The question says, which of the following could be the average i.e. there are probably many numbers that could be the average and one of them in included in this list.

We also know that 2a – b = 3c
Since right hand side has a 3, we know that 2a - b is divisible by 3. The easiest way to make it divisible by 3 is to make both a and b divisible by 3 which makes their sum divisible by 3 as well. Of the given options, only 24 is divisible by 3 hence (E) must be the answer.

jlgdr · Answer

My way of solving 2a = 3c + b a = (3c + b)/2 So average (a + b)/2 We replace a and we get 3c+b/2, finally we get 6c + 3b = 3(2c+b) as the average So average must be a multiple of 3 and higher than 9 since numbers must be non zero integers Therefore only 12 fits the bill Hope it helps! Let me see that Kudos rain!!! Cheers J

jlgdr · Answer

Here's actually another way of solving. First since all integers are >=0 it will be impossible to get a negative number so A,B are out right off the bat

Now, we have that 2a - b = 3c and we need to find a+b / 2

So let's begin with answer choice E

We have that a+b = 24

If we add

2a - b = 3c
a + b = 24

We have 3a = 24 + 3c

Now 24 + 3c is always a multiple of 3 so this one stays

Let's try with D

2a - b = 3c
a+b = 10

3a = 3c+10

Now 3c+10 won't ever be a multiple of 3 so OUT

One final try, C

2a -b = 3c
a+b = 2

3a = 3c+2

Again, 3c + 2 will NEVER be a multiple of 3.

Therefore only answer choice that is valid is E

Gimme some freaking Kudos!!!!
Cheers
J

kinjiGC · Answer

Let N be the average
so a+b = 2N
2a -b = 3c
Adding both the equation
3a = 2N + 3c

As a,N,c are all integers, N has to be multiple of 3 to have the equation balanced. Hence 12 - Option E)

TimeTraveller · Answer

2a - b = 3c

2a - b - 3c = 0

3a + 3b + 3c = a + 4b + 6c

\frac{(a+b+c)}{3} = a + 4b + 6c

(a+b+c) = 3(a + 4b + 6c)

So, the average is a multiple of 3, hence Ans (E).

GMATinsight · Answer

We can always solve such questions by taking certain values for a, b and c keeping in mind that if average of a and b must be integers then both a and b must be either even or both must be odd
Also 2a - b must be a multiple of 3

Trying with Even Numbers first

Let, a=8, b=4 i.e. 2a-b = 16-4 = 12 = 3c i.e. c=4 i.e. Average of a and b = (8+4)/2 = 6

Obtained Average Relates with 12 so doubling every number in previous step

i.e. Let, a=16, b=8 i.e. 2a-b = 32-8 = 24 = 3c i.e. c=8 i.e. Average of a and b = (16+8)/2 =  12 SUCCESS

Answer: Option E

JeffTargetTestPrep · Answer

We can manipulate the first equation to read:

2a - 3c = b

Now let’s set up an expression for the average of a and b:

(a + b)/2 = ?

(a + 2a - 3c)/2 = ?

(3a - 3c)/2 = ?

3(a - c)/2 = ?

Since 3 is not divisible by 2, so a - c must be divisible by 2. Therefore, since (a - c)/2 must be an integer and 3(a - c)/2 must be a multiple of 3. The only multiple of 3 in the answer choices is E, 12; thus, E is the correct answer.

Answer: E

vallequalle · Answer

A little bit more rough method, but it worked in one minute:

We see that  2a - b   has  to be  divisible by   3  for c to be a non-zero integer.

So if we choose a number that is divisible by 3 as our average for a and b, we will  always  get a multiple of 3 as our "2a - b".

(For example take 3 as our average:
a = 4, b = 2, result will be 6, divisible by 3, check
a = 5, b = 1, result will be 9, divisible by 3, check
a = 9, b = -3, result will be 21, divisible by 3, check)

So lets look at the answer choices and only 12 is divisible by 3, hence it is 12 (E)

(a = 11, b = 13, result will be 9, divisible by 3, check)

Hope it helped at least someone as much as it helped me (Of course all those examples are not necessary to get to the result).

If 2a – b = 3c, where a, b, and c are non-zero integers

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If 2a – b = 3c, where a, b, and c are non-zero integers

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