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

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?

Tutors, sign in to answer this question.

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

x=amount invested in one account

y=amount invested in a second account

x+y=15,250

8.5%x+10%y=1411.75

0.085x+0.1y=1411.75

solve for x in x+y=15,250

x=15,250-y

substitute into the equation 0.085x+0.1y=1411.75

0.085(15,250-y)+0.1y=1411.75

1296.25-0.085y+0.1y=1411.75

1296.25+0.015y=1411.75

0.015y=115.5

y=115.5/0.015

y=7700

x+7700=15,250

x=7550

$7550 invested at 8.5 %

$7700 invested at 10 %

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...

I break it down for you . Always choose a variable for unknown ( what problem is asking you to find out)

X - amount invested in 8.5%

15250 - X = amount invested in 10%

Gain in 8.5%= x (0.085)

Gain in 10% account = ( 15250 - X ) 0.10

X ( 0.085) + ( 15200 - X ) =1411.75 Total gain in 2 accounts

Now by converting English statement of problem to algebraic expression , came up with a equation of

one unknown to solve, and I let you to solve.

Let x and y be the amounts invested in the two accounts. Then

x+y=15250

The total interest is

0.085*x + 0.1*y=1411.75

Solve the first equation for y and substitute into the second equation:

0.085*x + 0.1*(15250-x)=1411.75

Solve for x. After a little bit of algebra :) , you get

x=7550, y=7700

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.