Question

Katie invested a total of ​$8000​, part at 5% simple interest and part at 6% simple interest. At the end of 1​ year, the investments had earned ​$455 interest. How much was invested at each​ rate?

Answer 1

She invested "a" at 5% and "b" at 6%.

now, we know whatever amounts "a" and "b" are, they add up to 8000, thus a + b = 8000.

how much is 5% of a? well, (5/100) * a, or 0.05a.
how much is 6% of b? well, (6/100) * b, or 0.06b.

After a year, their combined interest, their yield, was 455, thus, we also know that 0.05a + 0.06b = 455.

$\bf \begin{cases} a+b=8000\implies \boxed{b}=8000-a\\ 0.05a+0.06b=455\\ ----------\\ 0.05a+0.06\left( \boxed{8000-a} \right)=455 \end{cases} \\\\\\ 0.05a-0.06a+480=455\implies -0.01a=-25\implies a=\cfrac{-25}{-0.01} \\\\\\ \boxed{a=2500}$

how much was invested at 6%?  well, b = 8000 - a.