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

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

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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Source: Beat The GMAT — Problem Solving | Sat Jul 11, 2020 11:48 pm

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

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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swerve: Legendary Member; Posts: 2545; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

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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Scott@TargetTestPrep: GMAT Instructor; Posts: 8156; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

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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[email protected]

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gentvenus: Senior | Next Rank: 100 Posts; Posts: 59; Joined: Tue Oct 01, 2019 5:49 am

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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