There was a good world-wide response to this question. James
Bollard of St. Peter's College, Australia, answered the first
question.Only the whole numbers 0 and 2 will have their sum equal
to their product.
The second question about the relationship between the numbers,
where one of the numbers is an integer, was successfully answered
by a number of people: Kate Lister and Katherine Taylor students at
The Emmbrook School,Wokingham,England; Ling Xiang Ning of Tao Nan
School, Singapore; Jonathan Bloxham of the Royal Grammar School,
Newcastle, England; and Joyce Fu and Sheila Luk both students at
the Mount School, York, England. They all come to the correct
solution that numbers of the form $n$ and $n/(n-1)$ will have sums
equal to their product. Joyce Fu and Sheila Luk also point out this
will be true for negative numbers.
Claire Kruithof and Cinde Lau of Madras College, Scotland, go
further to investigate and find other pairs of numbers for which
the same relationship holds.
Catherine Aitken and Elisabeth Brewster, also of Madras College,
found these two related pairs and another family of solutions
(where $x$ and $y$ are whole numbers and $y \geq x$):
Well done, Catherine and Elisabeth. Their proof is as
follows:
$$\begin{eqnarray} \\ \left(\frac{(y + 1)}{x} \right) +
\left(\frac{(y + 1)}{(y - x + 1)}\right) &=& \left(\frac{(y
+ 1)(y - x + 1 + x)}{x(y - x + 1)}\right) \\ &=&
\left(\frac{(y + 1)^2}{x(y - x + 1)}\right). \end{eqnarray}$$
To find other families of solutions like this you simply take two
algebraic fractions with the same numerator and any two
denominators that add up to give the numerator.