Question

Last year, Keiko had $20,000 to invest. She invested some of it in an account that paid %7 simple interest per year, and she invested the rest in an account that paid %5 simple interest per year. After one year, she received a total of $1,280 in interest. How much did she invest in each account?

Answer 1

Answer:

$P_2 = \ 6,000\\P_1 = \ 14,000$

Step-by-step explanation:

The formula of simple interest is:

$I = P_0rt$

Where I is the interest earned after t years

r is the interest rate

$P_0$ is the initial amount

We know that the investment was $20,000 in two accounts

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For the first account r = 0.07 per year.

Then the formula is:

$I_1 = P_1r_1t$

Where

$P_1$ is the initial amount in account 1 at a rate $r_1$ during t = 1 year

$I_1 = P_1(0.07)(1)\\\\I_1 = 0.07P_1$

For the second account r = 0.05 per year.

Then the formula is:

$I_2 = P_2r_2t$

Where

$P_2$ is the initial amount in account 2 at a rate $r_2$ during t = 1 year

Then

$I_2 = P_2(0.05)(1)\\\\I_2 = 0.05P_2$

We know that the final profit was I $1,280.

So

$I = I_1 + I_2=1,280$

Substituting the values $I_1$, $I_2$ and I we have:

$1,280 = 0.07P_1 + 0.05P_2$

As the total amount that was invested was $20,000 then

$P_0 = P_1 + P_2 = 20,000$

Then we multiply the second equation by -0.07 and add it to the first equation:

$0.07P_1 + 0.05P_2 = 1.280\\.\ \ \ \ \ \ \ \ +\\-0.07P_1 -0.07P_2 = -1400\\-------------$

$-0.02P_2 = -120\\\\P_2 = 6,000$

Then $P_1 = 14,000$