Question 1185242
A cistern can be filled by two pipes.
 let x = time required by the larger pipe alone
:
 The small pipe alone will take 24 minutes longer than the larger pipe to fill the cistern alone.
 (x+24) = time required by the smaller pipe
:
 The small pipe alone will take 32 minutes longer to fill the cistern alone than when the two pipes are operating together.
let t = time required when working together
then
t + 32 = x+24
t = x + 24 - 32
t = x - 8, time required when working together
:
Let the full cistern = 1
{{{(x-8)/x}}} + {{{(x-8)/(x+24)}}} = 1
multiply by x(x+24)
(x+24)(x-8) +  x(x-8) = x(x+24)
x^2 - 8x + 24x - 192 + x^2 - 8x = x^2 + 24x
Combine like terms on the left
x^2 + x^2 - x^2 - 8x + 24x - 8x - 24x - 192 = 0
x^2 - 16x - 192 = 0
a quadratic equation which we can factor
(x-24)(x+8) = 0
positive solution
x = 24 min, large pipe working alone
:
:
Check solution
small pipe alone: 24 + 24 = 48 min
working together: 24 - 8 = 16 min
{{{16/24}}} + {{{16/48}}} = 
{{{2/3}}} + {{{1/3}}} = 1