Strange mathematical riddle, from Dudeney's Amusements in Maths.

Solution 1:

You are doing fine so far. Each person's spending is the square of the numbers you quote. When Mrs. Robinson spent $6$s more than Mrs. Evans, that is $288$ farthings and $27^2-21^2=288.$ Now Mrs. Smith spent the most, so she bought $203$, Mrs. Jones spent about four times Mrs. Evans, getting about twice as much, so she bought $43$, leaving $69$ for Mrs. Brown. That means Bessie only bought $6$, so is Evans. Annie bought $48$ more than Mary, so Annie Brown bought $66$ and Mary Robinson $18$. Annie spent more than Emily, so Emily Jones bought $38$ leaving $202$ for Ada Smith.