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

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

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`