Question

a person collected $2,800 on a loan of $2,400 they made 7 years ago. if the person charged simple interest, what was the rate of interest?
the interest rate is \\(\square\\)%
(type an integer or decimal rounded to the nearest hundredth as needed.)

Explanation:

Step1: Recall Simple Interest Formula

The simple interest formula is $I = P \times r \times t$, where $I$ is the interest, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the time in years. We need to solve for $r$. Rearranging the formula gives $r=\frac{I}{P\times t}$.

Step2: Identify Values

From the problem, $I = 2800 - 2400 = 400$ (wait, no—wait, the total collected is $2800$, principal $P = 2400$, time $t = 7$ years. Wait, actually, the interest earned $I$ is the total collected minus principal? Wait, no—if they collected $2800$ on a loan of $2400$ over 7 years with simple interest, then the interest $I = 2800 - 2400 = 400$. So $I = 400$, $P = 2400$, $t = 7$.

Step3: Substitute into Formula

Substitute into $r=\frac{I}{P\times t}$: $r=\frac{400}{2400\times7}$. First calculate denominator: $2400\times7 = 16800$. Then $r=\frac{400}{16800}\approx0.0238095$. To convert to percentage, multiply by 100: $0.0238095\times100\approx2.38\%$. Wait, wait—wait, maybe I misread. Wait, the total collected is $2800$, principal is $2400$, so interest $I = 2800 - 2400 = 400$? Wait, no—maybe the total amount repaid is $2800$, so interest $I = 2800 - 2400 = 400$. Then $t = 7$ years. So $r=\frac{I}{P\times t}=\frac{400}{2400\times7}=\frac{400}{16800}\approx0.0238$, so $2.38\%$? Wait, let's recalculate: $400\div(2400\times7)=400\div16800\approx0.0238095$, times 100 is approximately $2.38\%$ (rounded to nearest hundredth of a percent). Wait, but maybe I made a mistake in $I$. Wait, maybe the collected amount is the total, so principal $P = 2400$, total amount $A = 2800$, so $I = A - P = 400$, $t = 7$. So formula $A = P(1 + rt)$ for simple interest? Wait, no, simple interest is $A = P + I = P + Prt = P(1 + rt)$. So $2800 = 2400(1 + 7r)$. Then divide both sides by 2400: $\frac{2800}{2400}=1 + 7r$. Simplify $\frac{2800}{2400}=\frac{7}{6}\approx1.1666667$. Then subtract 1: $0.1666667 = 7r$. Then $r=\frac{0.1666667}{7}\approx0.0238095$, same as before. So $r\approx2.38\%$ when converted to percentage (multiply by 100). Wait, but let's check again. Wait, maybe the problem is that the collected amount is the interest? No, the problem says "collected $2800$ on a loan of $2400$ they made 7 years ago". So the total amount received (principal + interest) is $2800$, principal $2400$, so interest $400$, time 7 years. So simple interest formula: $I = Prt$, so $r = I/(Pt) = 400/(2400*7) = 400/16800 ≈ 0.0238$, so $2.38\%$ (rounded to nearest hundredth).

Answer:

The interest rate is approximately $\boxed{2.38}\%$ (or if we did the calculation correctly, let's re-express: Wait, maybe I messed up the interest. Wait, maybe the $2800$ is the interest? No, that doesn't make sense. Wait, the problem says "collected $2800$ on a loan of $2400$ they made 7 years ago". So the loan was $2400$, and after 7 years, they collected $2800$ (so principal + interest = $2800$). So interest $I = 2800 - 2400 = 400$. Then $r = I/(Pt) = 400/(2400*7) = 400/16800 ≈ 0.0238095$, so $2.38\%$ when multiplied by 100. So the answer is approximately $2.38\%$.