Question

A person expends $240 in the purchase of wheat. If he had paid 20 cents a bushel less he could have obtained 100 bushels more for the same money. How many bushels did he buy?

Answer 1

He bought x bushels of wheat.
The price per bushel was y.

xy = 240

Ok so far?

Answer 2

Now use again x as the number of bushels he actually bought and y as the actual cost per bushel and write an equation for "If he had paid 20 cents a bushel less he could have obtained 100 bushels more for the same money."

Answer 3

Do you follow?

Answer 4

Yes. Continue please?

Answer 5

But I'm not sure how to form an equation for "If he had paid 20 cents a bushel less he could have obtained 100 bushels more for the same money."

Answer 6

Ok. Let's work on that together.

First, keep this in mind:
The number of bushels multiplied by the cost per bushel = the total price paid.

For example, if a bushel costs $1.25, and he bought 100 bushels,
then the total cost is
100 bushels * 1.25 dollars/bushel = $125

Answer 7

Yes, I got that.

Answer 8

Above we used x as the number of bushels.
y is the cost per bushel.
That is why xy (x multiplied by y) is the total price he paid.

Now let's look at what he could have done.
The price could have been $0.20 less per bushel.
If the price per bushel is y, how do you write 20 cents less than y as an expression using y?

Answer 9

y-0.2 ?

Answer 10

Correct.
If the price had been $0.20 less per bushel, he would have bought 100 more bushels.
He bought x bushels.
How do you write 100 more than x as an expression using x.

Answer 11

x+100

Answer 12

Correct again.

Answer 13

Now you have a new cost per bushel: y = 0.2,
and a new number of bushels: x + 100
The total cost remains $240.

Can you write an expression that shows the new cost per bushel times the new number of bushels equaling the total cost?

Answer 14

(y-0.2)(x+100)=240

Answer 15

Is the answer 300 bushels?

Answer 16

Correct.

That equation, together with

xy = 240,

forms a system of equations.
Now you need to solve the system of equations to find x and y.

Answer 17

Does x=300 and x= -400? Therefore, the answer is 300 because it has to be positive?

Answer 18

Let's see:

xy = 240

Solve for y:

y = 240/x

Substitute in other equation:

(240/x - 0.2)(x +100) = 240

240 + 24000/x - 0.2x - 20 = 240

24000/x - 0.2x - 20 = 0

24000 - 0.2x^2 - 20x = 0

0.2x^2 + 20x - 24000 = 0

x^2 + 100x - 120,000 = 0

(x + 400)(x - 300) = 0

x + 400 = 0 or x - 300 = 0

x = 400 or x = -300

You are correct.

Answer 19

Isn't -400 and 300?

Answer 20

We can check.
Since x = 300, then xy = 240, giving us
y = 240/x = 240/300 = 0.8

The original cost was $0.80 per bushel.

Answer 21

Yes, of course, I just copied it wrong.

Answer 22

Look at my next line where I used x = 300.

Answer 23

The original cost was $0.80 per bushel.
Then $0.80 * 300 = $240

Now do $0.20 less per bushel, and 100 bushels more:
$0.60 * 400 = $240

Therefore, our answer of 300 bushels is correct.

Answer 24

Great work!

Answer 25

Thank you!

Answer 26

You're welcome.

Answer 27

I'm not sure but my teacher said the answer is 120,000. Is there a possible way for that to happen? My teacher is going to go over it next week, but I hate waiting!

Answer 28

@mathstudent55

Answer 29

The answer to this problem is not 120,000.
If $240 bought 120,000 bushels, then each bushel cost $0.002, or 0.2 cents.
It would then be impossible to pay $0.20 less per bushel.
Now look at the answer we got.
The number of bushels bought was 300. Each bushel cost $240/300 = $0.80
Now reduce the price of each bushel by $0.20 to have a per bushel price of $0.60.
The same $240 now buys you 240/0.60 = 400 bushels, which is 100 more bushels than 300, just as the problem stated.