Two bookworms, Alpha and Beta, start eating at opposite ends of a 42

Question

Two bookworms, Alpha and Beta, start eating at opposite ends of a 42-volume encyclopedia set (numbered 1 to 42), each working directly towards the other. Alpha starts in vol 1 and Beta starts in vol 42. If each volume is the same thickness, where in the set will they be when they meet?

(1) Alpha takes 3 x days to eat through a vol and Beta takes 5 x days, where x > 0.
(2) Alpha eats two volumes more than Beta every 4 days.

(1) Alpha takes 3 x days to eat through a vol and Beta takes 5 x days, where x > 0.
(2) Alpha eats two volumes more than Beta every 4 days.

skovinsky · Accepted Answer

Hi!

This is a classic ratio question. Here's the big rule to remember:

if you multiple or divide the parts, you can solve the ratio; if you add or subtract, you can't.

We know the distance travelled before they worms meet (42), so we need either their individual speeds or their relative speeds.

(1) we know their relative rates (a:b = 3x:5x = 3:5), so we can figure out how much each will eat before they meet: sufficient.

(2) with addition or subtraction, you can't figure out the ratio. We can set up:

a:b = 4x+2:4x

but without knowing the value of x, there's no way to figure out where they'll meet.

For example, if worm a eats 2 volumes per day, then in 4 days worm a eats 8 volumes and worm b eats 6 volumes - our ratio is 8:6 = 4:3.

However, if worm a eats 1 volume per day, then in 4 days worm a eats 4 volumes and worm b eats 2 volumes - our ratio is 4:2 = 2:1.

Since we can get different ratios, (2) is insufficient.

(1) is sufficient, (2) isn't: choose (A).

chetan2u · Answer

A and B start from two corner towards each other and there are 42 volumes..
(1) Alpha takes 3 x days to eat through a vol and Beta takes 5 x days, where x > 0.
 Most important point is that when they meet both have spent equal time 
Let A finishes y volume, so B finishes 42-y volume..
Time taken by A = 3x*y=3xy, while time taken by B = 5x*(42-y)..
Thus 3xy=5x(42-y)......3y=210-5y...8y=210...y = 26.25
So B does 42-y=42-26.25
  You could also do straight calculations.. work done will be proportionate to their speed..  
so speed 3x:5x will mean the work done will be 5:3...
Thus, work done by A =  42*\frac{5}{8}=\frac{210}{8}=26.25 
Hence sufficient

Hope it helps

mastvita · Answer

Thanks for the Reply

arpitkansal · Answer

chetan2u  can you explain how 1st statement alone is sufficient?

patto · Answer

Would you please explain the way the kaplan instructor(Above) solved the problem? Thanks

Fdambro294 · Answer

The key to getting through this difficult problem is to understand Proportionality and Ratios.

If A starts at one end of the Volume of Books and B starts on the other end, you can picture the 2 worms as if they were 2 cars driving in opposite directions towards each other.

They leave at the same Time. Thus, when A meets B somewhere in the middle of the 42 books, each will have been traveling for the SAME TIME.

When the Time traveled is Constant, the Ratio of the Speeds traveled at is Directly Proportional to the Ratio of the Distance Covered.

Assume each book is N = "unit thickness" (i.e., Distance) of each book.

the Gap Distance between A and B is = 42(N)

Where in the set will A and B meet?

S1:

A takes 3x days to "move" though 1 book

B takes 5x days to "move" though 1 book.

Again, when A and B meet, they will have been traveling for the SAME Time.

Ratio of: (A's Speed) : (B's Speed) = (1 book / 3x days) : (1 book / 5x days) = (1/3) : (1/5)

---convert the Ratio into Integers by Multiplying by *15---

Ratio of: (A's Speed) : (B's Speed) = 5 : 3

---->given the SAME Time traveled by each, the Ratio of Speeds Directly Proportional to the Ratio of Distance Traveled when they MEET ---->

Ratio of: (A's Distance) : (B's Distance) = 5 : 3

when they meet: A will have covered a Distance of = (5) / (3 + 5) = (5/8)th of the entire Gap Distance between them, which is the 42 Volumes

this means A and B will MEET at: (5/8) * 42 Books ---- Away from Volume 1, where A starts from

S1 is Sufficient Alone

S2: A "covers" 2 books more than B every 4 days

Again, we are given a Constant Time over which A and B are compared. However, the statement does not provide us with a Constant Ratio of Speeds that we can use.

essentially, on a "per 4 day basis":

Speed of A = 2 + Speed of B

Without a Ratio of Speeds, we can not determine the Relative Distance each will travel with they eventually meet.

For example:

case 1: A eats 3 books every 4 days ---- B eats 1 book every 4 days

over a constant time period, the Ratio of Speeds is: (A's Speed) : (B's Speed) = 3 : 1

when they Meet, A will have covered a Distance of = (3/4) * 42 volumes away from the beginning of Volume 1

case 2: A eats 10 books every 4 days --- B eats 8 books every 4 days

over a constant time period, the Ratio of Speeds is: (A's Speed) : (B's Speed) = 10 : 8 = 5 : 4

when they Meet, A will have covered a Distance of = (5/9) * 42 volumes away from the beginning of Volume 1

since we can not find a unique answer for WHERE A and B will meet, S2 is NOT Sufficient

(A) S1 alone

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Two bookworms, Alpha and Beta, start eating at opposite ends of a 42

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