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

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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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.

+1 Kudos if you Like

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:))

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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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.

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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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!

Having denominators like 541 in this case is difficult to solve. So in such cases. you can round off 541 to the nearest number common to the numerator. Here, the numerator ends with 0 so the nearest number to 541 ending with 0 is 540.

Thus, 26000/540 would give you 48.14..... which gives you 48. And as you asked a query, my reply would be no guessing is done here. Just assumptions are made so that one can solve such big numbers easily!

Increase is a constant amount : (5.0)x cents a day
and decrease is constant amount of about (0.41....)x cents a days

ie. The rate of increase is approximately 12 times the rate of decrease (5/0.41 is around 12)

Now the deference that needs to be covered is 580-320 = 260 cents
This 260 cents will be covered 12 parts by the increase and 1 by decrease
a 12:1 split in 260 is 240 : 20
ie the increase (from 320) will account for 240 cents and decrease (from 580) will account from 20 cents

Hence the price will be matched at 320+240 or 580-20 = 560 (option E)

Distance between the two prices = 580 - 320 = 260 cents.

We can treat this as a RATE problem, with the prices of corn and wheat TRAVELING TOWARD EACH OTHER to meet somewhere between 320 cents and 580 cents.

Let x=1.
Rate of increase for corn = 5x = 5*1 = 5 cents per day.
Rate of decrease for wheat = \(x\sqrt{2} - x\) ≈ 1.4x - x = 0.4x = 0.4(1) = 0.4 cents per day

For the two prices to meet, they must WORK TOGETHER to cover the 260-cent distance between them.
When elements work together, ADD THEIR RATES:
Combined rate for corn and wheat = 5 + 0.4 = 5.4 cents per day.

Of every 5.4 cents covered by the two prices as they work together to meet, wheat covers 0.4 cents.
Implication:
Wheat will cover \(\frac{0.4}{5.4} = \frac{2}{27}\) of the 260-cent distance between corn and wheat.
Thus:
Distance covered by wheat = \(\frac{2}{27} * 260 ≈ 2 * \frac{261}{27} = 2 * \frac{29}{3} ≈ 2*10 = 20\) cents

Since the 580-cent wheat decreases by about 20 cents to meet the price of corn, we get:
580 - 20 = 560 cents


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