Which situation would yield the highest amount in the account? $\newline$ (A) Depositing $\$ 200$ in an account the earns $5 \%$ simple interest for two years. $\newline$ (B) Depositing $\$ 400$ in an account the earns $3 \%$ interest compounded annually $\newline$ (C) Depositing $\$ 200$ in an account the earns $5 \%$ interest compounded annually for two years. $\newline$ (D) Depositing $\$ 400$ in an account the earns $3 \%$ simple interest for two years.

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Grade 7

Evaluate two-variable equations: word problems

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Q. Which situation would yield the highest amount in the account?

\newline

(A) Depositing

\$ 200

in an account the earns

5 \%

simple interest for two years.

\newline

(B) Depositing

\$ 400

in an account the earns

3 \%

interest compounded annually

\newline

\$ 200

in an account the earns

5 \%

interest compounded annually for two years.

\newline

(D) Depositing

\$ 400

in an account the earns

3 \%

simple interest for two years.

Calculate Option A: Calculate the final amount for option A using the formula for simple interest. $\newline$ Simple interest formula: $A = P(1 + rt)$ $\newline$ Where $A$ is the amount of money accumulated after $n$ years, including interest. $\newline$ $P$ is the principal amount (the initial amount of money). $\newline$ $r$ is the annual interest rate (in decimal). $\newline$ $t$ is the time the money is invested for, in years. $\newline$ For option A: $\newline$ $P = \$(200)$ $\newline$ $r = 5\% = 0.05$ (as a decimal) $\newline$ $t = 2$ years $\newline$ $A = \$(200)(1 + 0.05 \times 2)$
Calculate Option B: Perform the calculation for option A. $\newline$ $A = (\$)200(1 + 0.10)$ $\newline$ $A = (\$)200(1.10)$ $\newline$ $A = (\$)220$
Calculate Option C: Calculate the final amount for option B using the formula for compound interest. $\newline$ Compound interest formula: $A = P(1 + r/n)^{(nt)}$ $\newline$ Where $A$ is the amount of money accumulated after $n$ years, including interest. $\newline$ $P$ is the principal amount. $\newline$ $r$ is the annual interest rate (in decimal). $\newline$ $n$ is the number of times that interest is compounded per year. $\newline$ $t$ is the time the money is invested for, in years. $\newline$ For option B: $\newline$ $P = \$400$ $\newline$ $r = 3\% = 0.03$ (as a decimal) $\newline$ $n = 1$ (compounded annually) $\newline$ $A$ $0$ years $\newline$ $A$ $1$
Calculate Option D: Perform the calculation for option B. $\newline$ $A = \$(400)(1 + 0.03)^2$ $\newline$ $A = \$(400)(1.03)^2$ $\newline$ $A = \$(400) \times 1.0609$ $\newline$ $A = \$424.36$
Compare Final Amounts: Calculate the final amount for option C using the compound interest formula. $\newline$ For option C: $\newline$ $P = \$200$ $\newline$ $r = 5\% = 0.05$ (as a decimal) $\newline$ $n = 1$ (compounded annually) $\newline$ $t = 2$ years $\newline$ $A = \$200(1 + 0.05/1)^{(1*2)}$
Compare Final Amounts: Calculate the final amount for option C using the compound interest formula. $\newline$ For option C: $\newline$ $P = \$200$ $\newline$ $r = 5\% = 0.05$ (as a decimal) $\newline$ $n = 1$ (compounded annually) $\newline$ $t = 2$ years $\newline$ $A = \$200(1 + 0.05/1)^{(1*2)}$ Perform the calculation for option C. $\newline$ $A = \$200(1 + 0.05)^2$ $\newline$ $A = \$200(1.05)^2$ $\newline$ $A = \$200 \times 1.1025$ $\newline$ $A = \$220.50$
Compare Final Amounts: Calculate the final amount for option C using the compound interest formula. $\newline$ For option C: $\newline$ $P = \$200$ $\newline$ $r = 5\% = 0.05$ (as a decimal) $\newline$ $n = 1$ (compounded annually) $\newline$ $t = 2$ years $\newline$ $A = \$200(1 + 0.05/1)^{(1*2)}$ Perform the calculation for option C. $\newline$ $A = \$200(1 + 0.05)^2$ $\newline$ $A = \$200(1.05)^2$ $\newline$ $A = \$200 \times 1.1025$ $\newline$ $A = \$220.50$ Calculate the final amount for option D using the simple interest formula. $\newline$ For option D: $\newline$ $P = \$400$ $\newline$ $r = 5\% = 0.05$ $0$ (as a decimal) $\newline$ $t = 2$ years $\newline$ $r = 5\% = 0.05$ $2$
Compare Final Amounts: Calculate the final amount for option C using the compound interest formula. $\newline$ For option C: $\newline$ $P = \$200$ $\newline$ $r = 5\% = 0.05$ (as a decimal) $\newline$ $n = 1$ (compounded annually) $\newline$ $t = 2$ years $\newline$ $A = \$200(1 + 0.05/1)^{(1*2)}$ Perform the calculation for option C. $\newline$ $A = \$200(1 + 0.05)^2$ $\newline$ $A = \$200(1.05)^2$ $\newline$ $A = \$200 \times 1.1025$ $\newline$ $A = \$220.50$ Calculate the final amount for option D using the simple interest formula. $\newline$ For option D: $\newline$ $P = \$400$ $\newline$ $r = 5\% = 0.05$ $0$ (as a decimal) $\newline$ $t = 2$ years $\newline$ $r = 5\% = 0.05$ $2$ Perform the calculation for option D. $\newline$ $r = 5\% = 0.05$ $3$ $\newline$ $r = 5\% = 0.05$ $4$ $\newline$ $r = 5\% = 0.05$ $5$
Compare Final Amounts: Calculate the final amount for option C using the compound interest formula. $\newline$ For option C: $\newline$ $P = \$200$ $\newline$ $r = 5\% = 0.05$ (as a decimal) $\newline$ $n = 1$ (compounded annually) $\newline$ $t = 2$ years $\newline$ $A = \$200(1 + 0.05/1)^{(1*2)}$ Perform the calculation for option C. $\newline$ $A = \$200(1 + 0.05)^2$ $\newline$ $A = \$200(1.05)^2$ $\newline$ $A = \$200 \times 1.1025$ $\newline$ $A = \$220.50$ Calculate the final amount for option D using the simple interest formula. $\newline$ For option D: $\newline$ $P = \$400$ $\newline$ $r = 5\% = 0.05$ $0$ (as a decimal) $\newline$ $t = 2$ years $\newline$ $r = 5\% = 0.05$ $2$ Perform the calculation for option D. $\newline$ $r = 5\% = 0.05$ $3$ $\newline$ $r = 5\% = 0.05$ $4$ $\newline$ $r = 5\% = 0.05$ $5$ Compare the final amounts from all options to determine the highest amount. $\newline$ Option A: $r = 5\% = 0.05$ $6$ $\newline$ Option B: $r = 5\% = 0.05$ $7$ $\newline$ Option C: $r = 5\% = 0.05$ $8$ $\newline$ Option D: $r = 5\% = 0.05$ $9$ $\newline$ The highest amount is from option B: $r = 5\% = 0.05$ $7$ .