Question 178632: The sum, product, and quotient of two numbers are all equal. What are the numbers?
. Please help! I have tried every number combination I can think of and nothing seems to work!
Thanks!
Found 2 solutions by stanbon, Mathtut:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! The sum, product, and quotient of two numbers are all equal. What are the numbers?
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a+b = ab
a+b = a/b
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ab = a/b
ab^2 = a
b^2 = 1
b = 1 or b = -1
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If b = 1 then a*1 = a and that is true.
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If b = -1 then a*-1 = a/-1 and that is true
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Now, what about a+b
If a+b = ab and b = 1 then a = a which tells us nothing
If a+b = ab and b = -1 then a-1 = -a and 2a = 1 and a = 1/2
And does that hold for a+b = a/b ?
Substituting a = 1/2 and b = -1 you get:
(1/2) -1 = (1/2)/-1
and -1/2 = -1/2
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Conclusion; a = 1/2 and b = -1 satisfy the problem conditions.
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Cheers,
Stan H.
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Answer by Mathtut(3670) (Show Source):
You can put this solution on YOUR website! lets call our numbers x and y
:
Well, if we call the numbers x and y, then we know that the following
must be true:
x + y = x * y
x + y = x / y
x * y = x / y
What we really want to do is get one of these variables (x or y) by
itself on one side of an equation, because then we can substitute for
it in another equation. Let's look at the third equation:
x * y = x / y
x * y * y = x
y * y = x / x
y^2 = 1
y = 1 or -1
That narrows things down considerably. Now let's look at the first
equation. Since y must be 1 or -1, one of the following must be true:
1) x + 1 = x * 1
2) x - 1 = x * -1
The first one can't be true for any number, since one operation
changes the value of x and the other doesn't. But what about the
second equation? Can you find a number that makes it true?
:
x-1=-x
:
2x=1
:
x=1/2
so -1 and 1/2
:
1/2-1=-1/2
1/2(-1)=-1/2
(1/2)/-1=-1/2
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