# Arithmetic: simple interest question

Alice puts 3400 dollar on her bank account with interest rate 3,7%.

How much interest does she receive in the second year?

I answered: $3525.8\cdot1.037-3400\cdot1,037=130.5$, but this is the incorrect answer.

Can someone help me?

$$I(t) = P_0rt$$
where $P_0$ is the initial principal, $100r\%$ is the annual interest rate, and $t$ is the time measured in years. The initial principal is $\$3400and the interest rate is $$\frac{3.7\%}{100\%} = 0.037$$ Thus, the amount of interest earned in the second year is the difference between the amount earned after two years and the amount earned after one year. \begin{align*} I(2) - I(1) & = \3400(0.037)(2) - \$3400(0.037)(1)\\ & = \$3400(0.037)(2 - 1)\\ & = \$3400(0.037)(1)\\ & = \$125.80 \end{align*}
Observe that if we replaced $1$ by $n$ and $2$ by $n + 1$, we would obtain the same answer. Thus, provided no money is added to or removed from the account, the interest earned in any year would also be $\$125.80$. This is simple interest. If an amount$P$is invested at an annual simple interest rate of$r$, then every year the interest earned is just$Pr$. This is true regardless of which year it is, because simple interest is always computed on the original principal. In your case, the interest earned every year is $$\underbrace{3400}_{\textrm{principal }P}\cdot \underbrace{0.037}_{\textrm{rate }r} =\underbrace{125.80}_{\textrm{annual interest }Pr}$$ 3400•0.037=125.8 3400+125.8=3525.8 3525.8•0.037=130.45 That should help you. After one year there will be:$3400\cdot(1.037) = 3525.8$dollars on the account. This means that she will recive$3400\cdot(0.037) = 125.8\$ dollars in interest in the second year.