SOLUTION: Juma bought a certain number of orange for 36 dollars. If each orange had cost 0.50 dollar less, he could have bought six more oranges for the same amount of money. How many orange

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Question 934984: Juma bought a certain number of orange for 36 dollars. If each orange had cost 0.50 dollar less, he could have bought six more oranges for the same amount of money. How many oranges did he buy?
Found 2 solutions by TimothyLamb, josmiceli:
Answer by TimothyLamb(4379) About Me  (Show Source):
You can put this solution on YOUR website!
x = number of oranges bought at original price
y = original dollar price per orange
---
xy = 36
y = 36/x
(x + 6)(y - 0.50) = 36
---
xy - 0.50x + 6y - 3 = 36
xy - 0.50x + 6y = 39
x36/x - 0.50x + 6*36/x = 39
36 - 0.50x + 216/x = 39
216/x - 0.50x = 39 - 36
216/x - 0.50xx/x = 3
216 - 0.50xx = 3x
0.50xx + 3x - 216 = 0
xx + 6x - 432 = 0
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the above quadratic equation is in standard form, with a=1, b=6 and c=-432
---
to solve the quadratic equation, by using the quadratic formula, copy and paste this:
1 6 -432
into this solver: https://sooeet.com/math/quadratic-equation-solver.php
---
the quadratic has two real roots at:
---
x = 18
x = -24
---
the negative root doesn't fit the problem statement, so use the positive root:
---
answer:
x = number of oranges bought at original price = 18
y = 36/x
y = 36/18
y = original dollar price per orange = 2
---
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Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +n+ = the actual number of
oranges he bought
-----------------
The cost of each orange he bought
was +36%2Fn+
-----------------
They are asking you " what if each
orange had cost +36%2Fn+-+.5+?
-----------------------------
Now you can say:
+36+=+%28+36%2Fn+-+.5+%29%2A%28+n+%2B+6+%29+
+36+=+36+-+.5n+%2B+216%2Fn+-+3+
+-.5n+%2B+216%2Fn-+3+=+0+
Multiply both sides by +n+
+-.5n%5E2+-+3n+%2B+216+=+0+
Multiply both sides by +10+
+-5n%5E2+-+30n+%2B+2160+=+0+
+-n%5E2+-+6n+%2B+432+=+0+
------------------------
Use the quadratic formula:
+n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+-1+
+b+=+-6+
+c+=+432+
--------------
+n+=+%28-%28-6%29+%2B-+sqrt%28+%28-6%29%5E2-4%2A%28-1%29%2A432+%29%29%2F%282%2A%28-1%29%29+
+n+=+%28+6+%2B-+sqrt%28+36+%2B+1728+%29%2F%28-2%29%29+
+n+=+%28+6+%2B-+sqrt%28+1764+%29%2F%28-2%29%29+
+n+=+%28+6+-++42+%29+%2F+%28-2%29+
( use the minus square root to make +n+ positive )
+n+=+36%2F2+
+n+=+18+
Juma bought 18 oranges originally
------------------
check:
+36+=+%28+36%2Fn+-+.5+%29%2A%28+n+%2B+6+%29+
+36+=+%28+36%2F18+-+.5+%29%2A%28+18+%2B+6+%29+
+36+=+%28+2+-+.5+%29%2A24+
+36+=+1.5%2A24+
+36+=+36+
OK
Here's the plot of +f%28n%29+=+-n%5E2+-+6n+%2B+432+
to locate the roots
+graph%28+400%2C+400%2C+-26%2C+26%2C+-20%2C+500%2C+-x%5E2+-+6x+%2B+432+%29+
Looks like a root at +n=18+