find the reciprocal of (-12)/5*10/-3
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ALGEBRA WORD PROBLEM
Koony D. asked • 04/14/16
The sum of two numbers is 10. The sum of their reciprocals is 5/12. What are the numbers?
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Andrew M. answered • 04/14/16
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Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/Honors
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Note: A reciprocal is the number turned over...
The reciprocal of 2/3 = 3/2, and so forth
Since x = x/1, the reciprocal of x is 1/x....
The sum of 2 numbers is 10:
Equation 1) x + y = 10
The sum of their reciprocals is 5/12:
Equation 2) 1/x + 1/y = 5/12
From equation 1 we can solve for x in terms of y
(or vice versa...)
x = 10-y
Plug 10-y in place of x in equation 2 and solve for y
1/(10-y) + 1/y = 5/12
We need to find a common denominator on the left.
That would be y(10-y)
(y/y)(1/(10-y)) + [(10-y)/(10-y)](1/y)
= y/(y(10-y)) + (10-y)/(y(10-y))
= (y+10-y)/(y(10-y))
= 10/(10y-y2)
This is the left side of the equal sign combined
to a single fraction. It is still equal to 5/12
10/(10y-y2) = 5/12
Cross multiply:
10(12) = 5(10y-y2)
120 = 50y - 5y2
Move everything to the left side to create a quadratic by
adding 5y2 and subtracting 50y from both sides
5y2 - 50y +120 = 0
Note that each term in the polynomial is divisible by 5
so factor out a 5
5(y2 - 10y + 24) = 0
Divide both sides by 5
y2 - 10y + 24 = 0
The factors of 24 that add to -10 are (-6)(-4) so this factors to:
(y-6)(y-4) = 0
Thus either y-6 = 0 and y=6
OR
y-4 = 0 and y=4
If y = 4 then x=6
If y = 6 then x=4
Your numbers are 4, 6
Check Answer: x+y=10 yes
1/x + 1/y = 5/12
1/6 + 1/4 = 5/12
2/12 + 3/12 = 5/12
5/12 = 5/12
Step-by-step explanation:
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