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24 Jun 2014, 06:47
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D
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Difficulty:
95%
(hard)
Question Stats:
39% (03:08) correct
61%
(03:07)
wrong
based on 1446
sessions
History
Date
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Not Attempted Yet
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
A. 190
B. 188
C. 150
D. 148
E. 40
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24 Jun 2014, 06:48
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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
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
25 Jun 2014, 10:47
Kudos
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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.
-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.
