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Intern  Joined: 02 Nov 2010
Posts: 7
If n is an integer, the greatest possible value of the  [#permalink]

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5
27 00:00

Difficulty:   45% (medium)

Question Stats: 61% (01:18) correct 39% (01:37) wrong based on 1095 sessions

### HideShow timer Statistics If n is an integer, the greatest possible value of the expression: 12 - |32 - 7n| is

A. -20
B. 1
C. 8
D. 9
E. 12

I know to answer this correctly I can just plug in numbers. I was wondering is there any tricks to solving it faster.
Thanks.
Math Expert V
Joined: 02 Sep 2009
Posts: 56275
Re: Are there any tricks to solving this question  [#permalink]

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5
10
PASSINGGMAT wrote:
if n is an integer, the greatest possible value of the expression: 12 - |32 - 7n| is

A) -20
B) 1
C) 8
D) 9
E) 12

I know to answer this correctly I can just plug in numbers. I was wondering is there any tricks to solving it faster.
Thanks.

Algebraic approach:

The greatest possible value of the expression $$12-|32-7n|$$ will be for the least value of $$|32-7n|$$. Now, the least possible value of an absolute value is 0 --> $$|32-7n|=0$$ --> $$n=\frac{32}{7}=4\frac{4}{7}$$, but we are told that $$n$$ is an integer so the least value of $$|32-7n|$$ will be for $$n=5$$ (the closest integer value to $$4\frac{4}{7}$$) --> $$n=5$$ --> $$12-|32-7n|=12-3=9$$.

Similar question from OG: if-y-is-an-integer-then-the-least-possible-value-of-139867.html
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Math Expert V
Joined: 02 Sep 2009
Posts: 56275
Re: If n is an integer, the greatest possible value of the  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Director  S
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Re: If n is an integer, the greatest possible value of the  [#permalink]

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2
Let the greatest possible value be x.

1. For n>4, i.e., when the value within the modulus is negative, the modulus becomes, -(32-7n)
12- (- (32-7n))=x
12+32-7n=x
x=44-7n
so when n>4, x is greatest for n=5 and the value of x is 9.

2. For n<=4 , the modulus is 32-7n
12-(32-7n)=x
So when n<=4, the greatest value of x is 8

So , of the two, x=9 is the greater value.
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Re: If n is an integer, the greatest possible value of the  [#permalink]

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PASSINGGMAT wrote:
If n is an integer, the greatest possible value of the expression: 12 - |32 - 7n| is

A. -20
B. 1
C. 8
D. 9
E. 12

I know to answer this correctly I can just plug in numbers. I was wondering is there any tricks to solving it faster.
Thanks.

good one...so...we need to get the greatest possible value..we have an absolute value..thus, it will always be positive.
since n must be an integer, 7n can't be 32. it can be either 28, or 35. in case it's 35, then we have |-3|. in case it is 28, then |4|. since we have a minus sign, we would rather have |-3|, since 12-3=9, but 12-4=8. thus, the max value is 9.
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Location: United States
Concentration: Entrepreneurship, Marketing
GPA: 3.5
Re: If n is an integer, the greatest possible value of the  [#permalink]

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SravnaTestPrep wrote:
Let the greatest possible value be x.

1. For n>4, i.e., when the value within the modulus is negative, the modulus becomes, -(32-7n)
12- (- (32-7n))=x
12+32-7n=x
x=44-7n
so when n>4, x is greatest for n=5 and the value of x is 9.

2. For n<=4 , the modulus is 32-7n
12-(32-7n)=x
So when n<=4, the greatest value of x is 8

So , of the two, x=9 is the greater value.

x is greatest for n = 5; x is 9 , when n = 6; x is 2
so greatest x =9
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Intern  B
Joined: 12 May 2017
Posts: 13
Re: If n is an integer, the greatest possible value of the  [#permalink]

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My approach, no difficult algebra needed, but please correct me if i'm wrong. I'm still learning!:

First look at |32 - 7n|.

On the number line you would be at number 32 with 7n to the left en 7n to the right. The question asks for the greatest possible value of 12 - X.
X stands for |32 - 7n|. So you know X must be small in order for the formula to turn out big.

You know n is an integer, and you know X needs to be small, so:
32-7*4=4 (Let's try one more)
32-7*5=-3 (Bingo)

-3 it is. (Absolute value, so 3)

12-3=9. There you go.
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Re: If n is an integer, the greatest possible value of the  [#permalink]

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maximum value for the expression
12-mod(x)
to maximise the value of the expression x must be minimum
mod(x)=35-7x
x=4 mod x =4
x=5 mod x =3
x=6 mod x =7

Hence max value of the expression =9
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Intern  B
Joined: 16 Jul 2018
Posts: 2
Re: If n is an integer, the greatest possible value of the  [#permalink]

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Let's think of it like this -

n is an integer so 7n must also be an integer. Correct?

Now the question asks for the greatest value, for this the part in the modulus needs to be be the smallest value. (The smaller the number is from 12 the better)

Inside the modulus we have 2 numbers |32-7n|

So, the value of 7n should be as close to the number 32 as possible to get a small value.

What multiples of 7 is the closest to 32? --->>> You guessed it right it is 35. That means n=5, (since 7x5=35)

So now let's get to the simple math -

32-35=-3, Since there is modulus, it will only be 3

Next -> 12-3=9. Which is the greatest value of the expression

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Re: If n is an integer, the greatest possible value of the  [#permalink]

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PASSINGGMAT wrote:
If n is an integer, the greatest possible value of the expression: 12 - |32 - 7n| is

A. -20
B. 1
C. 8
D. 9
E. 12

So if n is an Integer, we can take values and observe that when n = 5 we are getting maximum value for the expression

12 - |32-35|
12-3
9

D
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Many of life's failures happen with people who do not realize how close they were to success when they gave up. Re: If n is an integer, the greatest possible value of the   [#permalink] 16 Feb 2019, 20:17
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