Question

A positive integer is 6 less than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is frac(9,7), then find the two integers.

Answer 1

1 and 7

Explanation:

A graphical solution is quick and easy.

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Let x represent the smaller number. Then the given relationships say ...

1/x + 2/(x+6) = 9/7

Multiplying by 7x(x+6), we have ...

7(x+6) +14x = 9x(x+6)

9x^2 +54x = 21x +42 . . . . eliminate parentheses, swap sides

9x^2 +33x = 42 . . . . . . . . subtract 21x

3x^2 +11 -14 = 0 . . . . . . . . . subtract 42 and divide by 3

(3x +14)(x -1) = 0 . . . . . . . . factor

Values of x that make this true are x = 1 and x = -14/3. Then for positive integer x=1, the other integer is x+6=7.

The integers of interest are 1 and 7.