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# If m^2 < 225 and n - m = -10, what is the sum f the smallest

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Math Expert
Joined: 02 Sep 2009
Posts: 59721
If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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24 Jun 2014, 06:47
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95% (hard)

Question Stats:

37% (03:06) correct 63% (03:11) wrong based on 792 sessions

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If $$m^2 < 225$$ and $$n - m = -10$$, what is the difference between the smallest possible integer value of $$3m + 2n$$ and the greatest possible integer value of $$3m + 2n$$?

A. -190
B. -188
C. -150
D. -148
E. -40

Kudos for a correct solution.

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Posts: 59721
If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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24 Jun 2014, 06:48
22
19
SOLUTION

If $$m^2 < 225$$ and $$n - m = -10$$, what is the difference between the smallest possible integer value of $$3m + 2n$$ and the greatest possible integer value of $$3m + 2n$$?

A. -190
B. -188
C. -150
D. -148
E. -40

This question is about algebraic manipulations with inequalities.

From $$n - m = -10$$ it follows that $$n=m-10$$. Thus, $$3m + 2n=3m+2(m-10)=5m-20$$. So, we need to find the difference between the smallest possible integer value of $$5m-20$$ and the greatest possible integer value of $$5m-20$$.

Now, lets' work on $$m^2 < 225$$:
Take the square root from both sides: $$|m|<15$$;
Get rid of the modulus sign: $$-15<m < 15$$;
Multiply all three parts by 5: $$-75< 5m < 75$$;
Subtract 20 from all three parts: $$-95< 5m -20< 55$$;

From $$-95< 5m -20< 55$$ it follows that the smallest possible integer value of $$5m-20$$ is -94 and the greatest possible integer value of $$5m-20$$ is 54.

Therefore, the difference is -94 - 54= -148.

Try NEW inequalities DS question.
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Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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25 Jun 2014, 10:47
7
1
I get -148.

-15 < m <15
Also, 3m+2n= 5m-20, since n=m-10
For m=-15, which isn't an option, we get 5m-20= -95. But the exact previous number -94 is an option since m doesn't have to be an intenger. So that is our minimum
Similarly, for m=+15, which is not an option, 5m-20= +55. But the exact previous number +54 is an option. So this is our maximum.

So min-max = -94 -54= -148.
##### General Discussion
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Joined: 19 Jan 2008
Posts: 16
Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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24 Jun 2014, 10:22
2
Hi,
i am getting -140 as the answer(none of the options:().
My calculations are as below:-
min value of m=-14,thus n=-24
max value of m=14,thus n=4

Smallest 3m+2n=3(-14)+2(-24)=-42-48=-90
Largest 3m+2n=3(14)+2(4)=50

Diff between smallest and largest=-90-50=-140

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Posts: 4
Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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24 Jun 2014, 11:00

m^2<225 => |m|<15

n-m=-10 => m=n+10

As far as it is not mentioned that "m" and "n" should be integers => |m|=14,5

Calculating max value:
14,5=n+10 => n=4,5
3m+2n = 43,5+9=52,5 =>52 (as the smallest integer value)

Calculating min value:
-14,5=n+10 => n=-24,5
3m+2n = -43,5-49=-92,5 => -93 (as the smallest integer value)

Difference "min-max" = -93-52=-145

Hence "C"
Math Expert
Joined: 02 Sep 2009
Posts: 59721
Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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24 Jun 2014, 11:36
ankushbassi wrote:
Hi,
i am getting -140 as the answer(none of the options:().
My calculations are as below:-
min value of m=-14,thus n=-24
max value of m=14,thus n=4

Smallest 3m+2n=3(-14)+2(-24)=-42-48=-90
Largest 3m+2n=3(14)+2(4)=50

Diff between smallest and largest=-90-50=-140

Solution for this question will be published by the end of the week.

By the way this question is a part of our NEW PROJECT.
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Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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25 Jun 2014, 02:51
Anyone else wants to try?
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Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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25 Jun 2014, 21:27
3

-94 - 54

= -148

$$m^2 < 225$$

m can be anything from -14.9 to + 14.9

Kindly refer chart below for detailed calculation:
Attachments

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Math Expert
Joined: 02 Sep 2009
Posts: 59721
Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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29 Jun 2014, 12:15
2
SOLUTION

If $$m^2 < 225$$ and $$n - m = -10$$, what is the difference between the smallest possible integer value of $$3m + 2n$$ and the greatest possible integer value of $$3m + 2n$$?

A. -190
B. -188
C. -150
D. -148
E. -40

This question is about algebraic manipulations with inequalities.

From $$n - m = -10$$ it follows that $$n=m-10$$. Thus, $$3m + 2n=3m+2(m-10)=5m-20$$. So, we need to find the difference between the smallest possible integer value of $$5m-20$$ and the greatest possible integer value of $$5m-20$$.

Now, lets' work on $$m^2 < 225$$:
Take the square root from both sides: $$|m|<15$$;
Get rid of the modulus sign: $$-15<m < 15$$;
Multiply all three parts by 5: $$-75< 5m < 75$$;
Subtract 20 from all three parts: $$-95< 5m -20< 55$$;

From $$-95< 5m -20< 55$$ it follows that the smallest possible integer value of $$5m-20$$ is -94 and the greatest possible integer value of $$5m-20$$ is 54.

Therefore, the difference is -94 - 54= -148.

Kudos points given to correct solutions above.

Try NEW inequalities DS question.
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Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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10 Jul 2014, 11:16
Bunuel wrote:
ankushbassi wrote:
Hi,
i am getting -140 as the answer(none of the options:().
My calculations are as below:-
min value of m=-14,thus n=-24
max value of m=14,thus n=4

Smallest 3m+2n=3(-14)+2(-24)=-42-48=-90
Largest 3m+2n=3(14)+2(4)=50

Diff between smallest and largest=-90-50=-140

Solution for this question will be published by the end of the week.

By the way this question is a part of our NEW PROJECT.

Hi Bunuel..
I did also the same way above mentioned and got -140. However I did understand the method which you have explained in the discussion but still want to clarify about the method which has been explained above...
Can you pls explain...whats wrong with above method???

Math Expert
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Posts: 59721
Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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10 Jul 2014, 11:21
2
GGMAT760 wrote:
Bunuel wrote:
ankushbassi wrote:
Hi,
i am getting -140 as the answer(none of the options:().
My calculations are as below:-
min value of m=-14,thus n=-24
max value of m=14,thus n=4

Smallest 3m+2n=3(-14)+2(-24)=-42-48=-90
Largest 3m+2n=3(14)+2(4)=50

Diff between smallest and largest=-90-50=-140

Solution for this question will be published by the end of the week.

By the way this question is a part of our NEW PROJECT.

Hi Bunuel..
I did also the same way above mentioned and got -140. However I did understand the method which you have explained in the discussion but still want to clarify about the method which has been explained above...
Can you pls explain...whats wrong with above method???

We are NOT told that m is an integer, hence from -15<m<15 saying that the minimum value of m is -14 and the maximum value of m is 14 is wrong.
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Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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14 Aug 2014, 23:30
its D -148

I did it in a following way

here n - m = -10

m = n+10

then 3m +2n = 3(n+10)+2n = 5n+30

so we need to findout max and minimum values of 5n+30

m^2 <225

put m= n+10
(n+10)^2 <225

solving this equation becomes
n^2 +20n -125 <0

(n+25)(n-5) <0

so here we have three ranges

n <-25
-25 <n <5
n>5

taking n>5 it becomes >0 so sequence will be +-+

now n will be between -25 <n<5

taking n= 4.9 . 5n+30 will have value 54.5 so integer value is 54
taking n = -24.9. 5n+30 will become -124.5+30 = -94.5 = -94(integer value)

so difference is 54 - (-94) = 148
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Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest  [#permalink]

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27 Jan 2019, 02:58
This is my approach:

as $$m^2 < 225$$
so $$-15<m<15$$
m can take any value from -15 to 15 (note m is not necessarily an integer so m can take any decimal value as well)
no $$n-m=-10$$
as by looking at answer options i can say that all options are quite apart so i would put minimum value of $$m=15$$ and maximum value of $$m=15$$ and the resulting answer would be slightly more than required answer value.
for $$m=15$$, $$n=-25$$
for $$m=-15$$, $$n=5$$
so minimum value of $$3m+2n=3(-15)+2(-25)=-95$$
maximum value of $$3m+2n=3(15)+2(5)=55$$
Difference=$$-95-50=-150$$

So the answer would be slightly less than $$-150$$ and the correct option is D
Re: If m^2 < 225 and n - m = -10, what is the sum f the smallest   [#permalink] 27 Jan 2019, 02:58
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