If m^2

Question

If  m^2 < 225  and  n - m = -10 , what is the positive 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

Bunuel · Accepted Answer

Official Solution:

If  m^2 < 225  and  n - m = -10 , what is the positive 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 requires us to perform algebraic manipulations involving inequalities.

Given that  n - m = -10 , we can deduce that  n = m - 10 . Consequently,  3m + 2n = 3m + 2(m - 10) = 5m - 20 . Our goal is to find the positive difference between the smallest possible  integer value  of  5m - 20  and the greatest possible  integer value  of  5m - 20 . It's important to note that we are not given any information about whether  m  and  n  are integers. Keep this in mind when solving the problem.

Next, let's work with the inequality  m^2 < 225 :

Take the square root of 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 positive difference between the smallest and greatest integer values is  54 - (-94) = 54 + 94 = 148 .

Answer: D

gn1991 · Answer

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.

ankushbassi · Answer

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

Can you please help ,what is wrong in my calculations.

Bunuel · Answer

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 .

PareshGmat · Answer

Answer = D = -148

-94 - 54

= -148

m^2 < 225

m can be anything from -14.9 to + 14.9

Kindly refer chart below for detailed calculation:

GGMAT760 · Answer

Hi Bunuel..
I 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???

Thanks in advance

Bunuel · Answer

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.

PathFinder007 · Answer

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

HasnainAfxal · Answer

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

CRACKGMATNUT · Answer

I am getting -149 with m=14.9 and while moving from 14.9 to 14.95 it's getting close to -150.

Can anyone explain why not -150 ?

Posted from my mobile device

MBAB123 · Answer

Hey, you cannot take m=14.9 as that will give us n=4.9 (for calculating the largest value). 3m + 2n = 14.9*3+4.9*2= 54.7, which is  not an integer . There for the max value that you can take is 14.8 and when you try it you will get the correct answer. 
P.S the question explicitly mentioned smallest possible  integer  value and largest possible  integer  value.

Hope this helps!

Kalpit1212 · Answer

=-15 -25 -95<3m+2n<55 Max Value = 54 Min =-94 Difference = -148

Regor60 · Answer

With the limitation on m^2, m must be:

-15<m<15

Substituting n-m = -10 into 3m + 2n:

= 5m - 20

Since m isn't restricted to integers but the desired result is, minimizing/maximizing m means a 5 in the denominator of m, meaning a maximum of 4 in the numerator:

m = -14 4/5, 14 4/5

Substituting into the target equation:

5(-74/5) - 20 = -94
5(74/5) - 20 = 54

54 - (-94) = 148

Posted from my mobile device

Kinshook · Answer

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 ?

m^2 - 225 < 0
(m-15)(m+15) < 0
-15 < m < 15

n = m - 10

-25 < n = m - 10 < 5

-45 < 3m < 45
-50 < 2n < 10
-45 - 50 = -95 < 3m + 2n < 45+10 = 55

The smallest possible integer value of 3m + 2n = -94
The largest possible integer value of 3m + 2n = 54

The difference between the smallest possible integer value of  3m + 2n  and the greatest possible integer value of  3m + 2n  = -94 - 54 = -148

IMO D

akshatv45 · Answer

M is between -15 and 15, n = m-10. So 3m+2n = 5(m-4), m max can be 14.8 where as min can be -14.8. Just take the difference after this.

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

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