How do you find distinct digits (a), (b), (c), (d), (e) such that the following are in increasing order: #-"(a)".26#, #-22/3#, #-"(b)"50%#, #-4.15#, #-4 1/"(c)"#, #0#, #5.1 xx 10^(-"(d)"#, #3/1000#, #0."(e)"#, #60.2%# ?

1 Answer
George C.
Sep 18, 2017

There are #8# possible solutions, as below...

Explanation:

Given:

#-color(blue)("(a)").26#, #-22/3#, #-color(blue)("(b)")50%#, #-4.15#, #-4 1/color(blue)("(c)")#, #0#, #5.1 xx 10^(-color(blue)("(d)")#, #3/1000#, #0color(black)(.)color(blue)("(e)")#, #60.2%#

Note that:

#-22/3 = -7 1/3 = -7.bar(3)#

Hence:

#color(blue)("(a)")# is #8# or #9#

#color(blue)("(b)")# is #5# or #6#

Note that #1/7 = 0.bar(142857)# so #4 1/7 < 4.15# and:

#color(blue)("(c)")# is #7#, #8# or #9#

Note that any number from #5.1 xx 10^(color(blue)(-9))# to #5.1 xx 10^(color(blue)(-5)) < 3/1000#

So:

#color(blue)("(d)")# is #5#, #6#, #7#, #8# or #9#

Finally:

#color(blue)("(e)")# is #5# or #6#

Putting it together:

#color(blue)("(b)")# and #color(blue)("(c)")# use up #5# and #6# in either order, leaving values #7#, #8# and #9#.

So possible solutions in increasing lexicographical order are:

#((color(blue)("(a)"), color(blue)("(b)"), color(blue)("(c)"), color(blue)("(d)"), color(blue)("(e)")), (8, 5, 7, 9, 6), (8, 5, 9, 7, 6), (8, 6, 7, 9, 5), (8, 6, 9, 7, 5), (9, 5, 7, 8, 6), (9, 5, 8, 7, 6), (9, 6, 7, 8, 5), (9, 6, 8, 7, 5))#

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