Tuesday, May 8, 2012

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