Question

A positive integer is 4 less than another. If 3 times the reciprocal of the smaller integer is subtracted from the reciprocal of the larger integer, then the result is −67. Find all pairs of integers that satisfy this condition.

Answer 1

(0.0147, 4.0147) and (-4.04, 0.04)

Explanation:

Let the smaller number be x

Let the larger number be y

If a positive integer is 4 less than another, then;

x = y - 4 ...1

If 3 times the reciprocal of the smaller integer is subtracted from the reciprocal of the larger integer and result is -67, then;

1/y - 3(1/x) = -67

1/y -3/x = -67 ...2

Substitute 1 into 2

1/y - 3/y-4 = -67

y-4-3y/y(y-4) = -67

-4-2y = -67(y^2-4y)

- 4-2y = -67y^2 +268y

-67y^2 +268y+2y + 4 = 0

-67y^2 +270y + 4 =0

67y^2 -270y - 4 = 0

Factorize

-270±√270² - 4(-4)(67)/2(67)

= -270±√72,900+1072/134

= -270±271.97/134

= -270 -271.97/134 and -270 +271.97/134

= -541.97/134 and 1.97/134

x = 0.0147 and -4.04

y = x+4

y = 0.0147+4

y = 4.0147

If x = -4.04

y = -4.04 + 4

y = 0.04

Hence the pair of values are (0.0147, 4.0147) and (-4.04, 0.04)