Question

A woman invests a total of $20,000 in two accounts, one paying 2.5% and the other paying 8% simple interest per year. Her annual interest is $830. How much did she invest at each rate?

Answer 1

To find out how much was invested in each account, we set up a system of equations based on the total investment of $20,000 and the total annual interest of $830, solving for the amounts invested at 2.5% and 8% simple interest rates.

Step-by-step explanation:

To solve the problem of determining how much a woman invested in each account, we can set up a system of linear equations. We know the total investment is $20,000, which is divided into two parts: one part at 2.5% simple interest and the other at 8% simple interest. The total annual interest from both accounts is $830.

Let's denote the amount invested at 2.5% as x and the amount invested at 8% as y. The two key equations that represent this situation are:

1. x + y = $20,000
2. x × 0.025 + y × 0.08 = $830

From the first equation, we can express y as y = $20,000 - x. We can then substitute this expression for y in the second equation to find the value of x which represents the amount invested at 2.5% interest.

After solving the system, we will know how much was invested at each interest rate to satisfy the condition that the total interest earned in a year is $830.

Answer 2

Let's have a variable for each rate
x = account 1 (2.5% rate of interest)
y = account 2 (8% rate of interest)

The formulas we need are:
0.025x + 0.08y = 830 (amount of interest paid)
x + y = 20 000 (combined amount invested)

To solve we will use substitution (isolate for one variable and plugging into the other formula):
x + y = 20 000
x = 20 000-y

0.025x + 0.08y = 830
0.025(20 000-y) + 0.08y = 830
500-0.025y+0.08y=830
500+0.055y=830
0.055y=330
y=6000

Now that we know y (the 2nd account) = $6000 we need to find x. We do this by using the previous formula.
x = 20 000-y
x=20 000 - 6000
x=14000

She invested $14 000 at a 2.5% interest rate and $6 000 at a 8% interest rate.

*Notes:
the percentages were converted into decimal form
you check your answer by plugging the numbers back into the 2 formulas we made