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04 Nov 2022, 10:34
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If \(m\) is a positive number and \(n\) is a negative number, and \(|m| > |n|\), then which of the following has the greatest value ?
A. \(|\frac{m - n}{n}|\)
B. \(|\frac{m - n}{m}|\)
C. \(|\frac{m + n}{m - n}|\)
D. \(|\frac{m + n}{n}|\)
E. \(|\frac{m + n}{m}|\)
A. \(|\frac{m - n}{n}|\)
B. \(|\frac{m - n}{m}|\)
C. \(|\frac{m + n}{m - n}|\)
D. \(|\frac{m + n}{n}|\)
E. \(|\frac{m + n}{m}|\)
04 Nov 2022, 10:34
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Official Solution:
If \(m\) is a positive number and \(n\) is a negative number, and \(|m| > |n|\), then which of the following has the greatest value ?
A. \(|\frac{m - n}{n}|\)
B. \(|\frac{m - n}{m}|\)
C. \(|\frac{m + n}{m - n}|\)
D. \(|\frac{m + n}{n}|\)
E. \(|\frac{m + n}{m}|\)
Probably the easiest wat to solve this problem would be plug the values. Let \(m=2\) and \(n=-1\) (this satisfies conditions given in the stem), then:
A. \(|\frac{m - n}{n}|=|\frac{2 - (-1)}{-1}|=|-3|=3\).
B. \(|\frac{m - n}{m}|=|\frac{2 - (-1)}{2}|=|1.5|=1.5\).
C. \(|\frac{m + n}{m - n}|=|\frac{2 + (-1}{2 - (-1)}|=|\frac{1}{3}|=\frac{1}{3}\)
D. \(|\frac{m + n}{n}|=|\frac{2 + (-1)}{-1}|=|-1|=1\).
E. \(|\frac{m + n}{m}|=|\frac{2 + (-1)}{2}|=|\frac{1}{2}|=\frac{1}{2}\)
As we can see the answer is option A.
Still, to test our absolute value skills, let's also solve the question using absolute value properties. Let's establish two points:
(i) \(|m - n| > |m + n|\). (this is because it's given that \(m\) is a positive number and \(n\) is a negative number. For example, \((|5 - (-1)| = 6) > (|5 + (-1)| = 4)\)).
(ii) \(|\frac{a}{b}|=\frac{|a|}{|b|}|\).
So:
A. \(|\frac{m - n}{n}|=\frac{|m - n|}{|n|}\)
B. \(|\frac{m - n}{m}|=\frac{|m - n|}{|m|}\)
C. \(|\frac{m + n}{m - n}|=\frac{|m + n|}{|m - n|}\)
D. \(|\frac{m + n}{n}|=\frac{|m + n|}{|n|}\)
E. \(|\frac{m + n}{m}|=\frac{|m + n|}{|m|}\)
A and B have the same numerator so let's compare these two options first (both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). Since given that \(|m| > |n|\), the A > B.
C, D and E have the same numerator, so let's compare these three options next (again, both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). The denominator of D is less than that of E (it's given that \(|m| > |n|\)) and since also given that \(m\) is a positive number and \(n\) is a negative number, the denominator of D is also less than that of C.
So, we are left to compare A and D. The numerator of A is greater than that of D (check (i) above) and their denominators are equal, so A > D. Therefore, A has the greatest value.
Answer: A
If \(m\) is a positive number and \(n\) is a negative number, and \(|m| > |n|\), then which of the following has the greatest value ?
A. \(|\frac{m - n}{n}|\)
B. \(|\frac{m - n}{m}|\)
C. \(|\frac{m + n}{m - n}|\)
D. \(|\frac{m + n}{n}|\)
E. \(|\frac{m + n}{m}|\)
Probably the easiest wat to solve this problem would be plug the values. Let \(m=2\) and \(n=-1\) (this satisfies conditions given in the stem), then:
A. \(|\frac{m - n}{n}|=|\frac{2 - (-1)}{-1}|=|-3|=3\).
B. \(|\frac{m - n}{m}|=|\frac{2 - (-1)}{2}|=|1.5|=1.5\).
C. \(|\frac{m + n}{m - n}|=|\frac{2 + (-1}{2 - (-1)}|=|\frac{1}{3}|=\frac{1}{3}\)
D. \(|\frac{m + n}{n}|=|\frac{2 + (-1)}{-1}|=|-1|=1\).
E. \(|\frac{m + n}{m}|=|\frac{2 + (-1)}{2}|=|\frac{1}{2}|=\frac{1}{2}\)
As we can see the answer is option A.
Still, to test our absolute value skills, let's also solve the question using absolute value properties. Let's establish two points:
(i) \(|m - n| > |m + n|\). (this is because it's given that \(m\) is a positive number and \(n\) is a negative number. For example, \((|5 - (-1)| = 6) > (|5 + (-1)| = 4)\)).
(ii) \(|\frac{a}{b}|=\frac{|a|}{|b|}|\).
So:
A. \(|\frac{m - n}{n}|=\frac{|m - n|}{|n|}\)
B. \(|\frac{m - n}{m}|=\frac{|m - n|}{|m|}\)
C. \(|\frac{m + n}{m - n}|=\frac{|m + n|}{|m - n|}\)
D. \(|\frac{m + n}{n}|=\frac{|m + n|}{|n|}\)
E. \(|\frac{m + n}{m}|=\frac{|m + n|}{|m|}\)
A and B have the same numerator so let's compare these two options first (both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). Since given that \(|m| > |n|\), the A > B.
C, D and E have the same numerator, so let's compare these three options next (again, both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). The denominator of D is less than that of E (it's given that \(|m| > |n|\)) and since also given that \(m\) is a positive number and \(n\) is a negative number, the denominator of D is also less than that of C.
So, we are left to compare A and D. The numerator of A is greater than that of D (check (i) above) and their denominators are equal, so A > D. Therefore, A has the greatest value.
Answer: A
