Question

Phyllis invested 4000 dollars, a portion earing simple interest rate of 5 per year and the rest earning a rate of 7 percent per year. After one year the total interest earned on these investments was 2280 dollars. How much money did she invest at each rate?

Answer 1

Let:

P1 = Investment 1

P2 = Investment 2

r1 = Interest rate 1 = 5% = 0.05

r2 = Interest rate 2 = 7% = 0.07

t = time (in years) = 1

The simple interest is given by:

$I=Prt$

So:

Let:

I1 = Interest 1

I2 = Interest 2

$\begin{gathered} I1=P1\cdot r1\cdot t \\ I1=P1\cdot0.05\cdot1 \\ I1=0.05P1 \\ ------------ \\ I2=P2\cdot r2\cdot t \\ I2=P2\cdot0.07\cdot1 \\ I2=0.07P2 \end{gathered}$

From the problem we know:

$P1+P2=4000$

Also, we know:

$\begin{gathered} I1+I2=2280 \\ so\colon \\ 2280=0.05P1+0.07P2 \end{gathered}$

So, let:

$\begin{gathered} P1+P2=40000_{\text{ }}(1) \\ 0.05P1+0.07P2=2280_{\text{ }}(2) \end{gathered}$

From (1) solve for P1:

$P1=40000-P2_{\text{ }}(3)$

replace (3) into (2):

$\begin{gathered} 0.05(40000-P2)+0.07P2=2280 \\ 2000-0.05P2+0.07P2=2280 \\ 2000+0.02P2=2280 \\ 0.02P2=2280-2000 \\ 0.02P2=280 \\ P2=\frac{280}{0.02} \\ P2=14000 \end{gathered}$

Replace the value of P2 into (3):

$\begin{gathered} P1=40000-14000 \\ P1=26000 \end{gathered}$