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The price of a bushel of corn is currently $3.20, and the price of a

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The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 11 Sep 2008, 03:49
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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 \(x\sqrt{2} - x\) 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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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 19 Nov 2013, 03:11
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dancinggeometry wrote:
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 \(\sqrt{2}*x-x\) 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


Note that we are not asked in how many days prices will cost the same.

Let \(y\) be the # of days when these two bushels will have the same price.

First let's simplify the formula given for the rate of decrease of the price of wheat: \(\sqrt{2}*x-x=1.41x-x=0.41x\), this means that the price of wheat decreases by \(0.41x\) cents per day, in \(y\) days it'll decrease by \(0.41xy\) cents;

As price of corn increases \(5x\) cents per day, in \(y\) days it'll will increase by \(5xy\) cents;

Set the equation: \(320+5xy=580-0.41xy\), solve for \(xy\) --> \(xy=48\);

The cost of a bushel of corn in \(y\) days (the # of days when these two bushels will have the same price) will be \(320+5xy=320+5*48=560\) or $5.6.

Answer: E.
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 30 Apr 2014, 03:09
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I Tried using Time/Rate approach:-

Initial Price Difference = 5.80 - 3.20 = 2.60
Price of Corn increasing by 5x
Price of Wheat decreasing by x(1.4)-x = .4x

Since both the quantities are moving towards reducing the Price Gap hence:-
Relative Increase = 5x +.4x
Let T be the Time by which GAP is filled so,
2.6 = T(5.4x) -> T= (2.6)/5.4X

Final Price = 3.20 + 5X * T -> 3.20 + 5*2.6/5.4 = 5.6

Answer E.

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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 11 Sep 2008, 10:16
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dancinggeometry wrote:
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 x(2^1/2) - 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


Lets y be the number of days until prices are equal.

The end price of corn = 3.2 + 5xy
The end price of wheat = 5.8 - x(1.4-1) = 5.8 - 0.4xy

The two end prices are equal -> 3.2 +5xy = 5.8 - 0.4xy
From here we get that 5.4xy = 2.6, or y = 2.6/5.4x or 26/54x

Replace y in first equation and you get 3.2 + 5x(26/54x), cross the x and you get approximately 3.2 + 2.4 = $5.6
Check price for 2nd equation = 5.8 - 0.4x(26/54X), cros x and you get 5.8 - 0.2 = $5.6


I got to y = 2.6/5.4X via a number of simple methods and it took me 4-5 mins to realize this is all I needed:))
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 18 Mar 2015, 05:59
Bunnel,

Could you please help here: how do you do the below the fastest way?

320+5xy=580-0.41xy, solve for xy --> xy=48; (Especially the last step 260/5.41)

Thanks in advance,
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 19 Mar 2015, 05:49
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Adilmalih84 wrote:
Bunnel,

Could you please help here: how do you do the below the fastest way?

320+5xy=580-0.41xy, solve for xy --> xy=48; (Especially the last step 260/5.41)

Thanks in advance,
AM


320 + 5xy = 580 - 0.41xy;

5xy + 0.41xy = 580 - 320;

5.41xy = 260

541/100*xy = 260

xy = 260*100/541 = ~48.
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 20 Mar 2016, 01:49
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I am not sure if this will be effective for all. Since the numbers are very small, direct decrease and increase of price will solve the problem quickly. But this is trail and error method. if not able to figure out in 30 seconds, should go ahead with the standard method.

Corn increase by 5x and wheat by x (2^1/2)-x which effectively is .4x

lets say x = .1, then corn increase by .5 and wheat decrease by .04

corn Wheat
day 0 3.2 5.8
day 1 3.7 5.76
day 2 4.2 5.72
day 3 4.7 5.68
day 4 5.2 5.64
day 5 5.7 5.6

So the price will match up in the range 5.6 to 5.64. Looking at options 5.6 is the answer.
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 06 Mar 2018, 10:58
dancinggeometry wrote:
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 x(2^1/2) - 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



hi Bunuel, niks18 :)

i have no idea if approached the question correctly, but I got correct answer :) 9i decoded this info x(2^1/2) - x as 2^1/2 = \(\sqrt{2}\) = 1.4 hence x(2^1/2) - x = 1.4x -1.4

here is it :) please advice if it is correct solution.

\(3,20 + 1.6x = 5.80 -1.4x -1.4\)

\(1.6x + 1.4x = 5.80 - 3.20-1.4\)

\(3x = 1.2\)

\(x = 2.5\)

\(3.20+2.50 = 5.70\)

so i chose \(E\)
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 06 Mar 2018, 11:07
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dave13 wrote:
hi Bunuel, niks18 :)

i have no idea if approached the question correctly, but I got correct answer :) 9i decoded this info x(2^1/2) - x as 2^1/2 = \(\sqrt{2}\) = 1.4 hence x(2^1/2) - x = 1.4x -1.4

here is it :) please advice if it is correct solution.

\(3,20 + 1.6x = 5.80 -1.4x -1.4\)

\(1.6x + 1.4x = 5.80 - 3.20-1.4\)

\(3x = 1.2\)

\(x = 2.5\)

\(3.20+2.50 = 5.70\)

so i chose \(E\)


Hey dave13 ,

I think you did something wrong here.

Look at the highlighted text above.

How can you get \sqrt{2}x - x = 1.4x - 1.4 ? It should be 1.4x - x

Please do the question again. Try to convert dollars to cents (make same units first) and then approach.

All the best :)
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Re: The price of a bushel of corn is currently $3.20, and the price of a  [#permalink]

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New post 21 Jul 2018, 06:28
Bunuel I'm a little confused by this last line:

541/100*xy = 260

xy = 260*100/541 = ~48.

Did you just divide 26,000 by 541 in your head or something? I'm confused how people are getting that ballpark 48 number. Guess and check? Thanks!
Re: The price of a bushel of corn is currently $3.20, and the price of a &nbs [#permalink] 21 Jul 2018, 06:28
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