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algebra 1 - reciprocals











































One number is 3 times another. If the difference of the reciprocals of the two numbers is 12 find the numbers

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Mar 16, 2011
Algebra 1 - Reciprocals
by: Staff

The question:

One number is 3 times another. If the difference of the reciprocals of the two numbers is 12 find the numbers


The answer:

One number is 3 times another. If the difference of the reciprocals of the two numbers is 12 find the numbers

I?m going to make a point of the definitions because there are actually two sets of solutions, depending on which number is first and which number is second.

x = 1st number (called "one number" in the problem description)

y = 2nd number (called "another number" in the problem description)

One number is 3 times another.

x = 3*y

If the difference of the reciprocals of the two numbers is 12 find the numbers.

The reciprocal of x is 1/x.

The reciprocal of y is 1/y.

The problem does not state whether 1/x - 1/y = 12, or 1/y - 1/x = 12.

The answer will be different, depending on which equation is used.

I'm going to assume the following is true because that is the order the numbers are presented in the statement of the problem:

1/x - 1/y = 12


There are now two equations with two unknowns:

x = 3*y

1/x - 1/y = 12


First, solve for y.

To solve for y, substitute 3*y for x in the second equation.

1/(3*y) - 1/y = 12

Multiply each side of the equation by y

y*[1/(3*y) - 1/y] = y*12

y*1/(3*y) - y*1/y = 12y

(1/3)*(y/y) - 1*(y/y) = 12y

(1/3)*1 - 1*1 = 12y

(1/3) - 1 = 12y

-(2/3) = 12y

Divide each side of the equation by 12

-(2/3)/12 = 12y/12

-(2/3)/12 = y*(12/12)

-(2/3)*(1/12) = y*1

-(2/3)*(1/12) = y

-(2*1)/(3*12) = y

-2/36 = y

-1/18 = y

Solve for x by substituting -1/18 for y in the first equation.

x = 3y

x = 3(-1/18)

x = -3/18

x = -1/6

The final answer is: x = -1/6, y = -1/18

Both answers are negative.



However, if . . . . . . . . . you use 1/y - 1/x = 12 for the second equation, the answers will be positive:

x = 1/6 and y = 1/18






Thanks for writing.


Staff
www.solving-math-problems.com


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