Paul invests $15,250 in two different accounts. One pays an interest rate of 8.5% while the other account pays 10%. If he gains a total of $1411.75 annually, how much did he invest in each account?

Sarah, your question is about "translating" English into algebra symbols. There are simpler ways to "solve" this problem, but I want to break it down as far as possible, because that is what I think is giving you trouble. Let's take each sentence, and start labeling things as we go along.

"Paul invests $15,250 in two different accounts."

We're told there are 2 accounts, lets give them names "a" and "b". We also know that Paul invested a total of $15,250 in these accounts, and when we total things, that is adding, so using the labels for the accounts, a & b, we translate this into "a + b = $15,250"

"One [account] pays an interest rate of 8.5% while the other account pays 10%"

This sentence gives us the interest rates on the two accounts. This looks like a simple interest problem. The generic formula for simple interest is that the Interest equals Principal times Rate times Time (I = Prt). Let's create two equations, one for each account. Label these interest rates "r", and the rate on account a as r(a) = 8.5% and the rate on account b as r(b) = 10%; a and b are the Principal amounts in each account, I(a) as the interest paid on account a and I(b) the interest on account b, and use t for time in both equations. Feel free to use any labels you want, but I created something like this:

I(a) = a • r(a) • t

I(b) = b • r(b) • t

"If he gains a total of $1,411.75 annually, how much did he invest in each account?"

The total gain from an investment is the interest, so Paul's total gain is the interest he receives from account a and the interest he gets from account b, and as a total, we are adding again, so we translate the first clause as I(a) + I(b) = $1,411.75.

We also know that since we are given an annual amount of interest, that the time "t" is 1 year, i.e. t = 1.

So let's sum up. We know

a + b = $15,250

I(a) = a • r(a) • t

I(b) = b • r(b) • t

I(a) + I(b) = $1,411.75

r(a) = 8.5%

r(b) = 10%

t = 1

We have labelled 7 variables [a, b, I(a), r(a), t, I(b), & r(b)] and have seven equations, so it is solvable. But I would recommend keeping a + b = $15,250, and plugging all the rest of the information into

I(a) + I(b) = $1,411.75

I(a) + I(b) = $1,411.75

Plugging in for I(a) and I(b)

[a • r(a) • t] + [b • r(b) • t] = $1,411.75

When working with interest rates expressed as percentages, remember that you need to use the decimal form for the math to workout, so 8.5% is equal to 0.085 and 10% is equal to 0.1, so let's plug in r(a), r(b) and t into that equation

[a • 0.085 • 1] + [b • 0.1 • 1] = $1,411.75

Simplifying that equation produces

0.085 • a + 0.1 • b = $1,411.75

And returning to our first equation

a + b = $15,250

These are two fairly straightforward equations, with two unknowns (a & b), and they are the unknowns the question asks for (how much did he invest in each account).

I'll leave the equation for you to solve; you can use substitution or any other method you prefer. Feel free to ask a follow-up question or email me, and I'd be happy to walk you through solving two equation--two unknown problems in more detail. I hope this helps. John