number puzzle

[bobobalde111]

lend me �2 said jill.
i'll pay it back
but i havent got that much, jack replied checking his cash

i'll let you have what ive got if you guess how much it is.
if i added half the square of half of it in pence i would get an exact number of pounds

how much did he have

[rogerr]

�1

[Muffet]

�1.92

[bobobalde111]

how do you arrive at 1.92 i dont think �1 is correct

[Muffet]

Let x = number of pence. Then x + half the square of x/2 must come to some multiple of 100.
So x + (x squared/8) is a multiple of 100.
So 8x + x squared is a multiple of 800.
So x(x+8) is a multiple of 800
So x and x+8 must both be even numbers. Both must end in 0 or one must end in 00, and x is less than 200, as there was less than �2. They can't both end in 0, so one must end in 00. That means 92 x 100, 100 x 108, or 192 x 200. Only the last is a multiple of 800. So x = 192.

[dynamicduo]

�1.92 is correct
half of �1.92 is 96p Square this = 9216. Half this = 4608
Add 4608 to original 192 = 4800 which is a whole number of pounds (�48)

[crofter]

bob (Davie) The answer of �1.92 is correct and the method above is "spot on". Well done Muffet (and dynamic duo).

[rogerr]

oops I thought it was a "trick" question

[bobobalde111]

mant thanks for answers today
i wasnt doubting answer for a second
i just couldnt get it to work out

[Muffet]

The clever fellow was Jack, working that out in his head so quickly. I had to write it all down on bits of paper.