I'm trying to understand a simple math concept intuitively. When you increase a sum by some percentage, why is it that the numbers that compose that sum are all increased at that percentage, instead of increasing at a percentage in proportion to the sum itself?
Put another way, why is it that if we have $ 2 \cdot 5 = 10 $. And we decide to increase $10$ by $50\%$ and obtain $15$, the sequence of numbers that compose $10$ increase by $50\%$ as well. So $2$ then increases to $3$, which gives $15$ as the sum. Why is this the case, rather than dividing the $50\%$ by $5$ giving $10\%$, and incrementing the each number in the sequence by $10\%$ instead?
Of course, the latter is clearly incorrect, but I'm curious if anyone has several intuitive ways to more clearly understand this basic idea.
I am very confused by your question, but you seem to be having trouble understanding a couple of things.
$\qquad (1)$: When a quantity is increased by $50\%$, we end up with $150\%$ of the original quantity. More generally, when a quantity is increased by $x\%$, we end up with $(100+x)\%$ of the original quantity. The end result is that the initial quantity is multiplied by $\frac{100+x}{100}$.
Similarly, if we were decreasing something by $x\%$, the end result is that the initial quantity is multiplied by $\frac{100-x}{100}$.
$\qquad (2)$: Multiplication is commutative and associative, meaning we can carry it out in steps, at each step calculating the product of any two factors in the multiplication.
In your example, we have that $10$ is multiplied by $\frac{100+50}{100}=\frac{150}{100}=1.5$. Thus we have
$$15=1.5\times 10=1.5\times (2 \times 5)=(1.5\times 2)\times 5=3\times 5$$
We simply chose to carry out the multiplication $1.5\times 2$ before anything else, but we could very well have opted for $1.5\times 5$ and got that $15=2\times 7.5$, which is also true.
$\qquad(3)$: Multiplication distributes over addition, but not over multiplication. In other words, the rule $a\times(b+c)=(a\times b)+(a\times c)$ is true, but the rule $a\times(b\times c)=(a\times b) \times (a\times c)$ is false.
This means that, for instance, we can write
$$10=2+2+2+2+2\\ 10=5+5$$
so that
\begin{align} 15=1.5\times 10&=1.5\times(2+2+2+2+2)\\ &=(1.5\times 2)+(1.5\times 2)+(1.5\times 2)+(1.5\times 2)+(1.5\times 2)\\ &=3+3+3+3+3 \end{align}
and similarly
\begin{align} 15=1.5\times 10&=1.5\times(5+5)\\ &=(1.5\times 5)+(1.5\times 5)\\ &=7.5+7.5 \end{align}
but
\begin{align} 15=1.5\times 10&=1.5\times(2\times 5)\\ &\neq(1.5\times 2)\times(1.5\times 5)\\ &=3\times7.5=22.5 \end{align}