Question 180812
Eleven groups of people were checking into a hotel. Group one made up 1/4 of the total number of people. Group two made up 2/6 of the total number of people. Group three made up 1/8 the total number of people. Group four made up 1/12 the total number of people. Group five made up 1/32 the total number of people. Group six made up 1/24 the total number of people. Group seven made up 1/16 the total number of people. Group eight made up 1/64 the total number of people. Group nine made up 1/64 the total number of people. Group ten made up 1/48 the total number of people. Group eleven consisted of 240 people. What was the total number of rooms needed if each room could be occupied by just one person?
:
let x = total number rooms required
:
{{{1/4}}}x + {{{2/6}}}x + {{{1/8}}}x + {{{1/12}}}x + {{{1/32}}}x + {{{1/24}}}x + {{{1/16}}}x + {{{1/64}}}x + {{{1/64}}}x + {{{1/48}}}x + 240 = x
Find the least common multiple of 64 & 48
64: 2*2*2*2*2*2
48: 2*2*2*2*3
2*2*2*2*2*2*3 = 192, multiply equation by the LCM to get rid of the denominators
192*{{{1/4}}}x + 192*{{{2/6}}}x + 192*{{{1/8}}}x + 192*{{{1/12}}}x + 192*{{{1/32}}}x + 192*{{{1/24}}}x + 192*{{{1/16}}}x + 192*{{{1/64}}}x +
 192*{{{1/64}}}x + 192*{{{1/48}}}x + 192(240) = 192(x)
:
Results of canceling the denominators:
48x + 32(2x) + 24x + 16x + 6x + 8x + 12x + 3x + 3x + 4x + 46080 = 192x
48x + 64x + 24x + 16x + 6x + 8x + 12x + 3x + 3x + 4x + 46080 = 192x
188x + 46080
188x - 192x = -46080
-6x = -46080
x = {{{(-46080)/(-6)}}}
x = 7,680 rooms required
;
A lot of math here, check my work.