what type of math is this? I am a total newbie to the world of math and was interesting in learning.  I just finished my degree(non-math) and am going to study a few math books to see if it interests me to apply for something more quantitative but I want to study something interesting with interesting problems that won't bore me.  
I thought about it, and thought of the type of problems that intrest me.  One is predicting the future and the other is predicting the past.  Here's a problem that I think would be cool. Say you have a list
calories 89, 34, 67, 43, 54, 232, 623

and someone tells you that someone had a total of "6553" calories in a day.  What type of math would try to figure this out? Is it algebra? (by the way to get this question all I did was take each value above and multiple first one by 1, second one by two, etc.up to 7.)
 A: To cast the problem a little more clearly, you have a number of "weights", $w_1,\ldots,w_n$, in this case:
\begin{align*}
w_1 &= 89\\
w_2 &= 34\\
w_3 &= 67\\
w_4 &= 43\\
w_5 &= 54\\
w_6 &= 232\\
w_7 &= 623,
\end{align*}
and a "target total" $T$, in this case $T=6553$. You want to find nonnegative integers $a_1,\ldots,a_n$ such that
$$a_1w_1 + \cdots + a_nw_n = T.$$
In its broadest sense, this is an example of what is called a Diophantine equation (an equation in which we require the solutions to be nonnegative integers, or more generally rational numbers). They are studied in the branch of mathematics called Number theory.
A: For your question you need to find positive integer solutions to  $89a +34b +67c +43d+ 54d +232d+ 623e=6553$, $\{a,...,e\}$ are the number of items you eat of each to get 6553 calories. Problems like this where you need to find integer solutions are called linear Diophantine equations. But you can treat them as puzzles and try to use some hit and trial methods/computations. Sometimes, though not always, Wolfram alpha might help.
