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Tough time with this problem

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Tough time with this problem

by ksc1940 » Sat Jul 28, 2012 5:31 pm
This is a question from gmat math tough problems file. I read the explanation but still don't quite get it. If someone can walk me step-by-step, I would really appreciate it. Thanks.

The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of 5x cents per day while the price of wheat is decreasing at a constant rate of (square root of 2)(x)-x cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?

(A) $4.50
(B) $5.10
(C) $5.30
(D) $5.50
(E) $5.60
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by eagleeye » Sat Jul 28, 2012 6:18 pm
ksc1940 wrote:This is a question from gmat math tough problems file. I read the explanation but still don't quite get it. If someone can walk me step-by-step, I would really appreciate it. Thanks.

The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of 5x cents per day while the price of wheat is decreasing at a constant rate of (square root of 2)(x)-x cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?

(A) $4.50
(B) $5.10
(C) $5.30
(D) $5.50
(E) $5.60

Price of wheat starts off at 580 cents and keeps decreasing.
Price of corn starts at 320 cents and keeps increasing.
We need to find at which point wheat price equals corn price after x days.
Price of wheat after x days = 580-(sqrt(2)-1)*x cents
Price of corn after x days = 320 + 5*x cents
Equating the two:

320 +5x = 580-(sqrt(2)-1)x
Sqrt(2) = 1.4 (approx).
Then
320+5x = 580-0.4x
=> 5.4x = 580-320 = 260
=> x = 260/5.4= ~48 days

Then price of wheat = 320+5x = 320+ 5*48 = 320+ 240 = 560

E = $5.60 is correct.

Cheers :)
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by ksc1940 » Sat Jul 28, 2012 6:54 pm
eagleeye wrote:
ksc1940 wrote:This is a question from gmat math tough problems file. I read the explanation but still don't quite get it. If someone can walk me step-by-step, I would really appreciate it. Thanks.

The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of 5x cents per day while the price of wheat is decreasing at a constant rate of (square root of 2)(x)-x cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?

(A) $4.50
(B) $5.10
(C) $5.30
(D) $5.50
(E) $5.60

Price of wheat starts off at 580 cents and keeps decreasing.
Price of corn starts at 320 cents and keeps increasing.
We need to find at which point wheat price equals corn price after x days.
Price of wheat after x days = 580-(sqrt(2)-1)*x cents
Price of corn after x days = 320 + 5*x cents
Equating the two:

320 +5x = 580-(sqrt(2)-1)x
Sqrt(2) = 1.4 (approx).
Then
320+5x = 580-0.4x
=> 5.4x = 580-320 = 260
=> x = 260/5.4= ~48 days

Then price of wheat = 320+5x = 320+ 5*48 = 320+ 240 = 560

E = $5.60 is correct.

Cheers :)
Thanks a lot for this explanation! One quick question. Why did you simplify down to square root of 2 minus 1 times x?
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by eagleeye » Sat Jul 28, 2012 7:17 pm
ksc1940 wrote:
Thanks a lot for this explanation! One quick question. Why did you simplify down to square root of 2 minus 1 times x?
In my case, force of habit. It is not necessary to do it. What is important is to approximate sqrt(2) = 1.4
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by ksc1940 » Sun Jul 29, 2012 11:54 am
Sorry. One more question. How do we know that x=number of days? The question doesn't specify that.
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by eagleeye » Sun Jul 29, 2012 12:34 pm
ksc1940 wrote:Sorry. One more question. How do we know that x=number of days? The question doesn't specify that.
You're right, it's not explicit. I assumed x as the number of days only to simplify the calculation.

Otherwise, you would do it this way, still getting the same answer.
Let d be the number of days at which point wheat price equals corn price.

Price of wheat after d days = 580-(sqrt(2)-1)*dx cents
Price of corn after d days = 320 + 5*dx cents
Equating the two:

320 +5dx = 580-(sqrt(2)-1)dx
Sqrt(2) = 1.4 (approx).
Then
320+5dx = 580-0.4dx
=> 5.4dx = 580-320 = 260
=> d = 260/(5.4x)= ~48/x days

Then price of wheat = 320+5dx = 320+ 5*(48/x)*x = 320+ 240 = 560

E = $5.60 is correct.

Cheers!
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