Joan Arlington has twice as much money invested at​ 5% simple annual interest as she does at​ 4%. If her yearly income from the two investments is ​$280​, how much does she have at each​ rate?

Answer 1

We will use the simple interest formula: A=P(1+rt)

where: A=total amount accrued, P=principal, r=rate of investment, t=time (years, here).

r and t are given, and we know that the total amount accrued from both accounts minus the invested amounts (principals) will be $280.

So

1. A₁=P(1+0.04*1) simplified: A₁=P(1.04)
2. A₂=2P(1+0.05*1) simplified: A₂=2P(1.05)

because the amount invested in the account with interest rate of 5% is twice as much as the amount invested at 4% interest rate.

A₁+A₂-P-2P=280 because the total income is $280.

Substituting in equations 1 and 2 we get:

P(1.04)+2P(1.05)-P-2P=280

1.04P+2.1P-3P=280

3.14P-3P=280

0.14P=280

P=2000

So, the amount invested at 4% interest rate = P = $2,000

The amount invested at 5% interest rate = 2P = $4,000

Answer 2

a = 2b

0.05a + 0.04b = 280

0.05(2b) + 0.04b = 280

0.1b + 0.04b = 280

0.14b = 280

b = $2000

a = 2(2000)

a = $4000