answer to last week's MGRE Math Beast Challenge

Correct! The answer is indeed 14.286%. Strangely enough, MGRE's initial reasoning is rather tortured. Instead of following their own advice about solving a problem quickly through the plug-and-chug method, they went for the brute-force approach and solved the problem with variables. It's only at the end that the gurus discuss plug-and-chug as an afterthought. Here's what MGRE wrote:

When calculating a percent change “from (original) to (new),” be careful to use the ratio (change/original), not (change/new) or (new/original).

Create some variables:
x = attendance in 2010
y = attendance in 2011
z = attendance in 2012

In 2012, attendance was greater than in 2011, and even greater than it had been in 2010. So, x < y < z.

The question asks for the percent change from 2010 to 2011, or [(y - x)/x]•100%. This can be rewritten as [(y/x)-1]•100%, so what we really need to find is y/x.

We are given two percent changes from one year to another, but watch out! The “from (original)” year is different for each percent given.

“In 2012, attendance at an annual sporting event was 5% greater than it was in 2011”:
Note that “5% greater than” a number means “105% of” that number.
z = 105% of y
z = 1.05y

“In 2012, attendance at an annual sporting event was ... 20% greater than it was in 2010”:
Note that “20% greater than” a number means “120% of” that number.
z = 120% of x
z = 1.2x

To find y/x, we’ll first set both z equivalents equal:
z = 1.05y = 1.2x
y/x = 1.20/1.05

The answer is [(y/x) - 1]•100% = (1.14285714 - 1)•100% = 14.285714%. Rounded to the nearest 0.001%, the final answer is 14.286%.

Alternatively, we could pick numbers. A smart number for z would be a multiple of 120 and 105 (reflecting 5% and 20% increases from an easy base of 100).

z = (105)(120)
y = (100)(120)
x = (105)(100)

The answer is [(y/x) - 1]•100% = {[(100•120)/(105•100)] - 1}•100% = 14.285714%. Rounded to the nearest 0.001%, the final answer is 14.286%

For what it's worth, my own method took me 45 seconds with an on-screen calculator.


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this week's MGRE Math Beast Challenge

From here:

In 2012, attendance at an annual sporting event was 5% greater than it was in 2011 and 20% greater than it was in 2010. What was the percent increase in attendance from 2010 to 2011?

Give your answer to the nearest 0.001%.

Go to it! I'm no longer sick, so my answer will definitely appear in the comments sometime during the next 36 hours. As you can tell, this is a "grid-in" problem, i.e., you have to write the correct answer, not select an answer from multiple choices.


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answer to last week's MGRE Math Beast Challenge

I was still sick last week, so I failed to answer this rather complicated question. Here is MGRE's answer:

Rebecca began with 288 friends, evenly divided among 12 months. Thus, she had 24 friends with birthdays in each month.
Let’s make a simple chart:

[chart 1]

Now we’ll simply calculate and record all the changes. First, “the number of Rebecca’s friends with birthdays in the last quarter of the year increased by 25%.”

Thus, October, November, and December’s totals collectively increased by 25%, so from 3(24) = 72 to 72(1.25) = 90. We don’t actually know if each month increased by 25% (from 24 to 30) or if their total simply increased by 25% (for instance, maybe the entire increase of 18 occurred in one month, bringing that month’s total to 42, and the number of people with birthdays in the other two months remained at 24). The new total is 90 either way, and this question is ultimately about the total. However, one of the later constraints in this problem mentions “the month with the largest number of birthdays,” so let’s put the increase all in one month, as it might ultimately be the relevant month.

[chart 2]

Next, “the number of friends with birthdays in each month beginning with “J” increased by one-third.” To increase a number by one-third, multiply by one and one-third (this is faster than multiplying by one-third and then adding it back to the original): 24 (4/3) = 32

[chart 3]

Next, “the number of people with birthdays in February was increased by 12.5%.” Since 12.5% is 0.125, multiply by 1.125 to ADD 12.5% percent to the original number in one step: 24(1.125) = 27

[chart 4]

Next, “the number of people with birthdays in March became 166.6666...% of the new number of people with birthdays in February.”

166.6666...% of 27 is simply one hundred percent of the number, plus another two-thirds. Since 2/3 of 27 is 18, the new total for March is 45. Or, in the calculator: 27(1.666666666...) = 45. (Actually, putting this in the calculator will yield 44.9999999...., since you didn’t actually type in infinity 6’s. This is fine! The answer is 45.)

[chart 5]

Next, “the number of people with birthdays in April became five less than 75% of the new number of people with birthdays in February and March combined.”

February + March = 27 + 45 = 72
75% of 72 = 72(0.75) = 54.
We need the number 5 less than that: 54 – 5 = 49.

[chart 6]

Now, “the number of people with birthdays in May increased by 1, and the number of people with birthdays in August became one less than 20% greater than the new number of people with birthdays in May.”

May is now simply 24 + 1 = 25.
August is one less than 20% greater than 25. In the calculator: 25(1.2) = 30, then one less, or 29.

[chart 7]

Finally, “September’s total increased to 6% less than one more than the new total for the month with the largest number of birthdays.”

The month with the largest number of birthdays is April, with 49. Remember that even if the 25% increase in the total for the last quarter of the year occurred in a single month, that month (October in our chart) would only have 42 people.

One more than 49 is 50.

September’s total is 6% less than 50. To decrease a number by 6%, take 94% of it (this is faster than finding 6% and subtracting it from the original): 50(0.94) = 47

[chart 8]

To calculate the final answer, simply add the “AFTER” row of the chart:

32 + 27 + 45 + 49 + 25 + 32 + 32 + 29 + 47 + 42 + 24 + 24 = 408

The correct answer is B.

I don't have time to do it right now, but I'll be adding the charts later tonight.


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this week's Math Beast Challenge problem

From here:

Rebecca had 288 Facebook friends, and noticed that an equal number of these friends had birthdays in each of the twelve months of the year. Then, Rebecca approved many friend requests at once. After doing so, the number of Rebecca’s friends with birthdays in the last quarter of the year increased by 25%, the number of friends with birthdays in each month beginning with “J” increased by one-third, the number of people with birthdays in February was increased by 12.5%, the number of people with birthdays in March became 166.6666...% of the new number of people with birthdays in February, the number of people with birthdays in April became five less than 75% of the new number of people with birthdays in February and March combined, the number of people with birthdays in May increased by 1, and the number of people with birthdays in August became one less than 20% greater than the new number of people with birthdays in May. Finally, September’s total increased to 6% less than one more than the new total for the month with the largest number of birthdays. Assuming no one de-friended her, after approving all her friend requests, how many Facebook friends did Rebecca then have?

(A) 396

(B) 408

(C) 453

(D) 512

(E) 696

Good Lord. Given the time it takes to read this problem, you'd run out of time to do the rest of the Quant section if this were an actual GRE! Anyway, go to it! I'm still sick, so my answer may or may not appear in the comments section below.


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rock and roll! got last week's MGRE problem correct!

I admit I felt shaky about my solution to last week's MGRE Math Beast Challenge problem, but I turned out to be correct. Here's MGRE's explanation:

Let’s put the words into equation form:

(Total Income – Exclusion)(Tax Rate) = Income Tax.

The question asks about total income, so we’ll solve the equation for Total Income:

Total Income – Exclusion = [Income Tax / Tax Rate]
Total Income = [Income Tax / Tax Rate] + Exclusion

To maximize total income, we’ll minimize Tax Rate (smaller denominator→larger value) and maximize the Exclusion:

Maximum Total Income = [$8700/0.15] + $9800 = $58,000 + $9,800 = $67,800.

To minimize total income, we’ll maximize Tax Rate (larger denominator→smaller value) and minimize the Exclusion:

Minimum Total Income = [$8700/0.35] + $5200 = $24,857.14 + $5,200 = $30,057.14.

The correct answers are C, D, E, and F.

The letters C, D, E, and F correspond to the values I had selected. MGRE arrived at the exact same range that I had arrived at, too: roughly $30,057 at the bottom end, and $67,800 at the top end.

Triomphe!


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this week's MGRE Math Beast Challenge

From here:

For Jack, income tax is between 15 and 35 percent of total income after an “exclusion” amount has been subtracted (that is, Jack does not have to pay any income tax on the exclusion amount, only on the remainder of his total income). If the exclusion amount is between $5200 and $9800, and Jack’s income tax was $8700, which of the following could have been Jack’s total income?

(Choose all that are appropriate)

$13,000

$23,100

$33,200

$43,300

$53,400

$63,500

$73,600

Go to it! I'll leave my response in the comments.


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