Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

In the figure above, an unshaded circle with diameter x is surrounded by a shaded region with uniform width of 2.  For what values of x is the area of the unshaded circle greater than the area of the shaded region?

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Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

Two cards are drawn consecutively, without replacement, from a deck originally consisting of 4 red and 6 black cards.

Quantity A

The probability that the first card is red

Quantity B

The probability that the second card is red

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Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We’re not kidding! Check out our upcoming courses here.

Most people dislike absolute value, and inequalities can tie us up into knots. Put them together, and we can have some major headaches! Let’s test one out.

Set your timer for 1 minute and 15 seconds for this Quantitative Comparison problem and GO! Read more

Here are the free GRE events we’re holding this week.

12/5/12 – New York, NY – Free Trial Class – 6:30-9:30 PM

12/5/12 – Washington, D.C. – Free Trial Class – 6:30-9:30 PM

Looking for more free events? Check out our Free Events Listings Page.

Need a break from writing your grad school applications? Take a moment to catch up on some recent stories circulating around the grad school community.

Happy Friday!

Should You Ask a Teaching Assistant for a Recommendation Letter? (About.com Graduate School)

Trying to decide who to ask to write your grad school recommendation letters? Here’s why it’s probably not the best idea to turn to your undergraduate TAs.

Grad School Application Checklist: 10 Months Out (US News Education)

It’s never a good idea to wait until the last minute to get your graduate school applications together. Here is US New Education’s third installment of advice for completing your applications in a timely manner.

What I know now: Grad School (Jeremy Yoder) (The Molecular Ecologist)

Planning to attend grad school for science? Check out what one postdoctoral associate wishes he’d known to do (and what he’s glad he did) in graduate school.
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Many word problems seem to require us to write formulas in order to solve. Certain problems, though, qualify for a neat technique: Smart Numbers. We can actually pick our own real numbers and use them to solve!

Set your timer for 2 minutes for this Fill-In problem and GO! (© ManhattanPrep)

* Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate, Carrie, who earns twice as much as Lisa, spends ¼ of her monthly paycheck on rent and ½ on food. If the two women decide to donate the remainder of their money to charity each month, what fraction of their combined monthly income will they donate? (Assume all income in question is after taxes.)

 

(No answer choices given; this is a fill-in-the-blank)

 

We’ve got two women, Lisa and Carrie, and they each spend a certain proportion of income on rent and on food. Annoyingly, the fractions don’t have the same denominators; even more annoyingly, the two women don’t make the same amount of money. All of that will make an algebraic solution challenging.

Here’s what an algebraic solution would look like. Let’s call Lisa’s income x. She spends (3/8)x on rent and (5/12)x on food. Add these together:

(3x/8) + (5x/12) = (9x/24) + (10x/24) = 19x/24

Subtract from 100%, or x:

24x/24 “ 19x/24 = 5x/24

Lisa donates 5/24 of x, her income, to charity. What about Carrie?

Carrie’s income is equal to 2x (because she makes twice as much as Lisa). How much does she spend on rent and food?
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Recently, we discussed various strategies for translating word problems into math. Let’s put that knowledge to the test on a challenging problem from a category that everybody hates: Rates.

Set your timer for 2 minutes and GO! (© ManhattanPrep)

*  A bullet train leaves Kyoto for Tokyo traveling 240 miles per hour at 12 noon. Ten minutes later, a train leaves Tokyo for Kyoto traveling 160 miles per hour. If Tokyo and Kyoto are 300 miles apart, at what time will the trains pass each other?

(A) 12:40pm

(B) 12:49pm

(C) 12:55pm

(D) 1:00pm

(E) 1:05pm

One of the strategies we discussed in the translation article was make the situation real. Put yourself into the situation and imagine you’re the one doing whatever the problem is describing. That will help you to set things up cleanly and correctly.

So what’s going on in this particular situation? First, you’re the conductor on the Kyoto train. At noon, you pull out of the station (instantly and magically traveling 240 miles per hour from the very start!). The track is 300 miles long; after one hour, where are you?

After one hour, it’s 1pm and you’ve gone 240 miles, so you’re just 300 “ 240 = 60 miles from Tokyo.

Okay, now switch jobs. You’re the Tokyo train conductor and you leave Tokyo at 12:10pm. After one hour, where are you? You’re going 160 miles an hour, so after 1 hour, it’s 1:10pm and you’re 300 “ 160 = 140 miles from Tokyo.

By 1:10p, have the two trains passed each other? Definitely, because train K (for Kyoto) is even further towards Tokyo at that point. Now, make a guess: do you think that the trains had already passed each other by 1p? Think about it before you read the next paragraph.
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Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

A perfect square is an integer whose square root is an integer.

Quantity A

The average (arithmetic mean) of the first 100 positive perfect squares.

Quantity B

The median of the first 100 positive perfect squares.

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Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

Set A consists of 350 consecutive multiples of 2. Set B consists of 200 consecutive multiples of 3. The median of Set A is 199.5 greater than the median of Set B.

Quantity A

The 30th percentile of Set A

Quantity B

The 70th percentile of Set B

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Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

If , which of the following could be true about the reciprocals of a, b, c, and d?

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