Simple and compound interest

From the problem: principal $P=2000$, rate $r=4\%=0.04$, time $t=3$ years.

The formula for simple interest is $I=P\times r\times t$. Substitute the values: $I=2000\times 0.04\times 3$

$I=2000\times 0.04=80\text{ per year}$ $I=80\times 3=\$240$

The interest earned is $240. Note that the question asks for interest earned, not the total amount.

The question asks for interest earned, which is $I=Prt$, not the total amount $A=P+I$.

**$2480\ast \ast wouldbethetotalamountifinterestwere$480, but $480$ itself is already too large (see below). This option confuses total amount with interest.

**$2240\ast \ast =$2000 + $240, which is the total amount, not just the interest earned. The question asks only for the interest.

**$480\ast \ast wouldbetheinterestiftheratewere8$2000 \times 0.04 \times 3 = 240$, not 480.

**$240\ast \ast iscorrect:$I = 2000 \times 0.04 \times 3 = $240$.

Compound interest is calculated on the principal plus all previously earned interest. Each year the amount is multiplied by $(1+r)$.

After year 1: $A=P(1+r)$. After year 2: $A=P(1+r)(1+r)=P(1+r{)}^2$. After $t$ years: $A=P(1+r{)}^t$.

The correct compound interest formula is $A=P(1+r{)}^t$. The exponent $t$ is what distinguishes it from simple interest.

We need the formula for compound interest, where interest is earned on previously accumulated interest, producing exponential growth.

$A=P(1+rt)$ is the formula for simple interest, not compound interest. It produces linear, not exponential, growth.

$A=P+Prt$ is identical to $P(1+rt)$ — just written differently. It is still the simple interest total amount formula.

$A=P\times r^t$ is incorrect because $r$ alone (e.g. $0.05^t$) would make the amount shrink rapidly rather than grow, and there is no $(1+r)$ factor.

$A=P(1+r{)}^t$ is the correct annual compound interest formula.