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The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more

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The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more

by VJesus12 » Sat Jul 11, 2020 11:48 pm

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The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more than \(a,\) then \(m\) must be

A. \(30\%\) less than \(b\)
B. \(42\frac67\%\) less than \(b\)
C. \(50\%\) less than \(b\)
D. \(66\frac23\%\) less than \(b\)
E. \(75\%\) less than \(b\)

[spoiler]OA=A[/spoiler]

Source: Manhattan GMAT
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Re: The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more

by Jay@ManhattanReview » Thu Jul 16, 2020 11:45 pm
VJesus12 wrote: ↑
Sat Jul 11, 2020 11:48 pm
 The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more than \(a,\) then \(m\) must be

A. \(30\%\) less than \(b\)
B. \(42\frac67\%\) less than \(b\)
C. \(50\%\) less than \(b\)
D. \(66\frac23\%\) less than \(b\)
E. \(75\%\) less than \(b\)

[spoiler]OA=A[/spoiler]

Source: Manhattan GMAT
Given that \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\), we have 2m = a + b.

Given that \(m\) is \(75\%\) more than \(a,\) we have m = 1.75a

So, from 2m = a + b, we have 2*1.75a = a + b => 2.5a = b => a = 2b/5

Again, from 2m = a + b, we have 2m = 2b/5 + b = 7b/5

m = 7b/10 = 0.7b = 70% of b. Or, m is 30% less than b.

Correct answer: A

Hope this helps!

-Jay
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Re: The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more

by swerve » Fri Jul 17, 2020 10:29 am
VJesus12 wrote: ↑
Sat Jul 11, 2020 11:48 pm
 The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more than \(a,\) then \(m\) must be

A. \(30\%\) less than \(b\)
B. \(42\frac67\%\) less than \(b\)
C. \(50\%\) less than \(b\)
D. \(66\frac23\%\) less than \(b\)
E. \(75\%\) less than \(b\)

[spoiler]OA=A[/spoiler]

Source: Manhattan GMAT
Here's my solution,
\begin{cases}
2m=a+b \\
a\left(1+\dfrac{75}{100}\right)=m
\end{cases}

Hence \(a=\dfrac{4}{7} \cdot m\)
So, \(b=\dfrac{10m}{7} \Longrightarrow m=\dfrac{7}{10} \cdot b = 0.7b\)

Hence \(30\) percent less than \(b\)

Therefore, A
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Re: The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more

by Scott@TargetTestPrep » Sun Jul 19, 2020 11:07 am
VJesus12 wrote: ↑
Sat Jul 11, 2020 11:48 pm
 The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more than \(a,\) then \(m\) must be

A. \(30\%\) less than \(b\)
B. \(42\frac67\%\) less than \(b\)
C. \(50\%\) less than \(b\)
D. \(66\frac23\%\) less than \(b\)
E. \(75\%\) less than \(b\)

[spoiler]OA=A[/spoiler]

Solution:

We can let a = 4, so m = 1.75 x 4 = 7 and b must be 10 so that (4 + 10)/2 = 7. In this case, we see that m (which is 7) is 70% of b (which is 10), or m is 30% less than b.

Alternate Solution:

We are given that m is 75% more than a, so we know that m = 1.75a.

We also use the formula for the average, obtaining:

(a + b)/2 = m

Substituting, we have:

(a + b)/2 = 1.75a

a + b = 3.5a

b = 2.5a

b/2.5 = a

Using the earlier equation m = 1.75a, we can substitute for a:

m = 1.75 * b/2.5

m = (7/10) * b

Since m is 70% of b, then m is 30% less than b.

Answer: A

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Re: The number \(m\) is the average (arithmetic mean) of the positive numbers \(a\) and \(b.\) If \(m\) is \(75\%\) more

by gentvenus » Sun Jul 19, 2020 12:45 pm
Let a = 4, so m = 1.75 x 4 = 7 and b must be 10 so that (4 + 10)/2 = 7
m (which is 7) is 70% of b (which is 10)
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