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?

Accepted Solution

Answer:[tex]P_2 = \$6,000\\P_1 = \$14,000[/tex]Step-by-step explanation:The formula of simple interest is:[tex]I = P_0rt[/tex]Where I is the interest earned after t yearsr is the interest rate[tex]P_0[/tex] is the initial amountWe know that the investment was $20,000 in two accounts_______________________________________________For the first account r = 0.07 per year.Then the formula is:[tex]I_1 = P_1r_1t[/tex]Where[tex]P_1[/tex] is the initial amount in account 1 at a rate [tex]r_1[/tex] during t = 1 year[tex]I_1 = P_1(0.07)(1)\\\\I_1 = 0.07P_1[/tex]For the second account r = 0.05 per year.Then the formula is:[tex]I_2 = P_2r_2t[/tex]Where[tex]P_2[/tex] is the initial amount in account 2 at a rate [tex]r_2[/tex] during t = 1 yearThen[tex]I_2 = P_2(0.05)(1)\\\\I_2 = 0.05P_2[/tex]We know that the final profit was I $1,280.So[tex]I = I_1 + I_2=1,280[/tex]Substituting the values [tex]I_1[/tex], [tex]I_2[/tex] and I we have:[tex]1,280 = 0.07P_1 + 0.05P_2[/tex]As the total amount that was invested was $20,000 then[tex]P_0 = P_1 + P_2 = 20,000[/tex]Then we multiply the second equation by -0.07 and add it to the first equation:[tex]0.07P_1 + 0.05P_2 = 1.280\\.\ \ \ \ \ \ \ \ +\\-0.07P_1 -0.07P_2 = -1400\\-------------[/tex][tex]-0.02P_2 = -120\\\\P_2 = 6,000[/tex]Then [tex]P_1 = 14,000[/tex]