Rational number between 9/10 and 3/4?

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Answer 1 · 2018-03-06T07:18:52

Some examples would be #33/40#, #9/11#, #4/5#, #6/7#, #7/8#, #8/9#

Between any two distinct rational numbers are an infinite number of other rational numbers, but here are some particular examples:

The arithmetic mean (i.e. average) of #9/10# and #3/4# is:

#1/2(9/10+3/4) = 1/2(18/20+15/20) = 1/2(33/20) = color(blue)(33/40)#

The harmonic mean of #9/10# and #3/4# is:

#2/(10/9+4/3) = 2/(10/9+12/9) = 2/(22/9) = color(blue)(9/11)#

Another rational number between #9/10# and #3/4# is:

#(9+3)/(10+4) = 12/14 = color(blue)(6/7)#

In fact, noting that both #3/4# and #9/10# are of the form #n/(n+1)#, we find:

#3/4 < color(blue)(4/5) < color(blue)(5/6) < color(blue)(6/7) < color(blue)(7/8) < 9/10#