A positive integer is twice another. The difference of the reciprocals of the two positive integers is 1/8. Find the two integers?

1 Answer
Ben
Jun 11, 2018

The integers are 4 and 8.

Explanation:

Let's look at the first statement, "A positive integer is twice another." If we wanted to represent the first integer as the letter #a# and the second as #b#, we could write this as a mathematical expression:

#a=2b#

The next statement is "The difference of the reciprocals of the two positive integers is 1/8" Understanding that the reciprocal of #arArr1/a# and #brArr1/b#, we can write this as follows:

#abs(1/a-1/b)=1/8#

I put the absolute sign in there because we just know the difference between the two, and not necessarily the order. Because we wrote the earlier expression #a=2b# we can write the expression in terms of #b#:

#abs(1/(2b)-1/b)=1/8#

we need to make the denominators the same for both fractions on the left hand side, so we will multiply #1/b# by #2/2#:

#abs(1/(2b)-2/(2b))=1/8#

#abs(-1/(2b))=1/8#

#1/(2b)=1/8#

#2b=8#

#color(green)(b=4)#

now that we know #b#, we can go back and solve for #a#:

#a=2b#

#a=2(4)#

#color(green)(a=8)#

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