Q: The reciprocal of two times a certain integer plus the reciprocal of 2 more than twice the integer equals?

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It is 2.

9

80

Call the unknown integer x. Then, from the problem statement, x - 3/x = 26/3, or:x2 - 3 = 26x/3; or x2 - (26/3)x - 3 = 0x = 9

2/x =16/20 16x = 40 x = 2.5

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2

It is 2.

??? explain better.

9

This question can be expressed algebraically as: (1/n) + (1/(2n)) + 2 = 23, (1/n) + (1/(2n)) =21, ((1+2)/(2n)) = 21, (3/(2n)) = 21, or 2n = (3/21), 2n = (1/7), so n = (1/14). This, by the way, is an elementary algebraic proof that the solution to the above relation is (1/14). Anyway, to answer the question, reread the question: "[What integer is such that] the reciprocal of the integer...". notice, the reciprocal of (1/14) is 14, which is the integer in question! ^_^

80

4

Call the unknown integer x. Then, from the problem statement, x - 3/x = 26/3, or:x2 - 3 = 26x/3; or x2 - (26/3)x - 3 = 0x = 9

A negative integer. Every time.

4

That has no integer solution. Three times an integer is another integer; if you subtract to integers, you get an integer again, not a fraction.

The equasion would be 10x 1/j = 5x1/9 (if j equals some number) so the answer would be j=18.