Question 590639
Let N represent the unknown number. 
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By definition the reciprocal of the number is {{{1/N}}}
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So the sum of the unknown number and its reciprocal is:
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{{{N + 1/N}}}
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The problem tells you that this sum is equal to 5/4 of the number meaning that it is equal to 5/4 times the number. This can be written as {{{(5/4)*N}}}
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So we can write the equation that says this as:
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{{{N + 1/N = (5/4)*N}}}
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We can now get rid of the N and the 4 in the denominator bys multiplying both sides of this equation (all terms) by 4*N as follows:
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{{{4*N*N + 4*N*(1/N) = 4*N*(5/4)*N}}}
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Cross out the denominators with the corresponding terms in the numerator as follows:
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{{{4*N*N + 4*cross(N)*(1/cross(N)) = cross(4)*N*(5/cross(4))*N}}}
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With these cancellations we are left with:
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{{{4*N*N + 4*1 = N*5*N}}}
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And doing the multiplications in each of the terms simplifies this equation to:
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{{{4N^2 + 4 = 5N^2}}}
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Collect the terms containing N on one side of this equation by subtracting {{{4N^2}}} from both sides of the equation. This subtraction eliminates the {{{4N^2}}} on the left side and the resulting equation is:
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{{{4 = 5N^2 - 4N^2}}}
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Doing the subtraction on the right side simplifies this to:
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{{{4 = N^2}}}
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Now solve for N by taking the square root of both sides:
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{{{sqrt(4) = sqrt(N^2)}}}
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After taking the square root of both sides you are left with two answers. Either:
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{{{+2 = N}}} or {{{-2 = N}}}
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The +2 and the -2 are possible values for N because in either case, if you square them to find N-squared you get +4 as the answer. So we now have that N, the unknown number, is either +2 or -2.
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Check these two answers by one at a time adding the number to its reciprocal and seeing if that results in 5/4 times the number as follows:
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{{{2 + 1/2 = (5/4)*2}}}
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or:
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{{{-2 + (-1/2) = (5/4)*(-2)}}}
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And if you work these two equations out you will find that in each of them, the right side is equal to the left side. This means that our two answers are correct. N can be either +2 or -2.
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Hope this helps you to understand the problem a little better.
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