Question 1207675
<pre>
There are more than 1 answer, because we are not given the number of sisters.

Starts with T.
Spends (3/8)T on food.  Has (5/8)T left.
Spends (1/3)(5/8)T = (5/24)T on clothes. Has (5/8)T-(5/24)T = (5/12)T left.
Shares equally with N sisters.
Since it says she shares equally with N sisters, that doesn't mean she gives
it all to them. She gives each of the siblings, including herself, (5/12)T/(N+1),
which is 208.

So we have the Diophantine equation:

{{{(expr(5/12)T)/(N+1)}}}{{{""=""}}}{{{208}}}

{{{expr(5/12)T}}}{{{""=""}}}{{{208(N+1)}}}

{{{5T}}}{{{""=""}}}{{{2496(N+1)}}}

{{{5T}}}{{{""=""}}}{{{2496N+2496}}}

Here is the way I was taught to solve first degree Diophantine equations.

Write each number in terms of its nearest multiple of the smallest coefficient
in absolute value. So write each 2496 as 2495+1 (because 2495 is the nearest
multiple of 5.

{{{5T}}}{{{""=""}}}{{{(2495+1)N+(2495+1)}}}
{{{5T}}}{{{""=""}}}{{{2495N+N+2495+1}}}

Divide through by 5

{{{T}}}{{{""=""}}}{{{499N+N/5+499+1/5}}}

Isolate the fraction terms:

{{{T-499N-499}}}{{{""=""}}}{{{N/5+1/5}}}

The left side is an integer, so the right side is also an integer.
Let that integer be A.

{{{N/5+1/5}}}{{{""=""}}}{{{A}}}

{{{N+1}}}{{{""=""}}}{{{5A}}}
{{{N}}}{{{""=""}}}{{{5A-1}}}, the number of sisters

Substitute in

{{{5T}}}{{{""=""}}}{{{2496N+2496}}}

{{{5T}}}{{{""=""}}}{{{2496(5A-1)+2496}}}

{{{5T}}}{{{""=""}}}{{{12480A-2496+2496}}}

{{{5T}}}{{{""=""}}}{{{12480A}}}

{{{T}}}{{{""=""}}}{{{2496A}}}

She spent {{{expr(5/24)T}}}{{{""=""}}}{{{expr(5/24)(2496A)}}}{{{""=""}}}{{{520A}}} on clothes

A   Number of sisters  Spent on clothes
          5A-1              520A
--------------------------------
1           4                520
2           9               1040
3          14               1560
4          19               2080

There are multiple answers.

Edwin</pre>