Question

divide 243 into 3 parts such that half of the first part ,one-third of the second part and one- fourth of the third part are equal

Answer 1

Number = 243

Let the number be divided into three parts a,b&c

a+b+c = 243

Now, according to the question ;
(1/2)a = (1/3)b = (1/4)c

-------------(multiply each by LCM of 2,3,4 = 12)
(12/2)a = (12/3)b = (12/4)c

6a = 4b = 3c
_______________________

a:b = 4/6 = 2:3

b:c = 3:4

a:b:c = 2:3:4
_______________________
Let the common ratio be x, then

2x + 3x + 4x = 243
9x = 243
x = 27
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Hence the three parts are ;
a = 2x = 2×27 = 54
b = 3x = 3×27 = 81
c = 4x = 4×27 = 108
==========================

Answer 2

let the numbers be x, y , z

So 1/2 x = 1/3 y = 1/4 z = k

x = 2k , y = 3k , z = 4k

Also,

x+y+z = 243

2k+3k+4k = 243

9k = 243

k = 27

So numbers are x = 54 , y = 81 , z = 108