What is the greatest number of rectangles with integer side lengths and perimeter 10 that can be cut from a piece of paper with width 24 and length 60?

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Answer 1 · 2016-08-22T20:20:58

#360#

If a rectangle has perimeter #10# then the sum of its length and width is #5#, giving two choices with integer sides:

The piece of paper has area #24xx60 = 1440#

This can be divided into #12xx20=240# rectangles with sides #2xx3#.

It can be divided into #24xx15 = 360# rectangles with sides #1xx4#

So the greatest number of rectangles is #360#.

Answer 2 · 2016-08-23T09:02:41

#360#

Calling #S = 60 xx 24 = 2^5 xx 3^2 xx 5 xx 1# the problem can be stated as

Determine

#max n in NN^+#

such that

#n le S/(a cdot b)#
#a + b = 5#
#{a, b} in {1,2,3,4}#

giving the feasible pairs

#{1,4},{2,3}# and the desired result is

#n = 1440/4 =360#