Simple Interest - Pre Algebra

1. Two siblings invest equal amounts of money. The first sibling invests for 3 years at 4% simple interest and earns $144 total interest. The second sibling invests at 6$180 total interest. How long did the second sibling invest?
   Answer: 2.5 years First, find the principal amount from the first sibling's investment. Using I = Prt: $144 = P × 0.04 × 3, so P = $144/0.12 = $1,200. Now for the second sibling with the same principal: $180 = $1,200 × 0.06 × t. Solving for t: t = $180/($1,200 × 0.06) = $180/$72 = 2.5 years. Choice A (2.0 years) would give interest of $1,200 × 0.06 × 2 = $144. Choice C (3.0 years) would give interest of $1,200 × 0.06 × 3 = $216. Choice D (3.5 years) would give interest of $1,200 × 0.06 × 3.5 = $252.
2. The simple interest formula can be rearranged to solve for time: $t = \frac{I}{Pr}$. A student uses this to find how long it takes for $500 invested at 3.5$87.50 in interest. She calculates $t = \frac{87.50}{500 \times 3.5} = \frac{87.50}{1750} = 0.05$. What error did she make and what is the correct answer?
   Answer: She forgot to convert the percentage to decimal form; the correct answer is 5 years When working with the simple interest formula $I = Prt$, you need to be careful about how percentages are represented in calculations. The formula requires the interest rate to be expressed as a decimal, not as a percentage. Let's work through this problem correctly. The student wants to find the time using $t = \frac{I}{Pr}$ where $I = 87.50$, $P = 500$, and the rate is 3.5%. The critical step is converting 3.5% to decimal form: $3.5\% = 0.035$. The correct calculation is: $t = \frac{87.50}{500 \times 0.035} = \frac{87.50}{17.5} = 5$ years. The student's error was using 3.5 instead of 0.035 for the rate, which made her denominator 100 times larger than it should be, resulting in an answer that was 100 times smaller. Looking at the wrong answers: (A) is incorrect because the student didn't calculate correctly, and 0.05 centuries would be 5 years anyway. (B) suggests using the wrong formula, but $t = \frac{I}{Pr}$ is the correct rearrangement of the simple interest formula. (C) claims an arithmetic error, but the division $\frac{87.50}{1750} = 0.05$ is mathematically correct—the error occurred earlier in the setup. Remember this key rule: whenever you see a percentage in any mathematical formula, convert it to decimal form by dividing by 100 before substituting. This is one of the most common errors in interest problems.
3. Three friends invest different amounts for the same length of time at the same interest rate. Sarah invests $P$ and earns $60 interest. Miguel invests$2P$and earns$120 interest. Chen invests $3P$ but only earns $$150 interest. What can you conclude?
   Answer: Chen invested for a shorter time period than the other two Using I = Prt, if all have the same rate r and time t, then interest should be proportional to principal. Sarah: $60 = P × r × t, so rt = 60/P. Miguel: $120 = 2P × r × t = 2P × (60/P) = $120 ✓. Chen should earn: 3P × (60/P) = $180, but only earned $150. Since Chen earned less than expected while using the same rate, he must have invested for less time. We can find Chen's time: $150 = 3P × r × t_Chen, so t_Chen = 150/(3P × r) = 150/(3P × 60/P) = 150/180 = 5/6 of the original time. Choice A assumes Chen made a calculation error. Choice B assumes a bank error. Choice D would mean Chen got a different rate, but the problem states same rate for all.
4. Devon wants to earn exactly $$180 in simple interest. He can invest at 4% annually for 6 years, or at 6% annually for a different time period, or at a different rate for 5 years. If all three options use the same principal amount, what is the relationship between the three principals?
   Answer: They must all be equal to $$750 Using I = Prt with I = $180 for all cases: Case 1: $180 = P × 0.04 × 6, so P = $180/(0.24) = $750. Case 2: $180 = P × 0.06 × t, so P = $180/(0.06t) = $3000/t. For P = $750, we need t = 4 years. Case 3: $180 = P × r × 5, so P = $180/(5r) = $36/r. For P = $750, we need r = 0.048 or 4.8%. Since the problem states all three options use the same principal amount, and we calculated P = $750 from the first option, all three must use P = $750. Choice B ($900) would require different interest calculations. Choice C is wrong because the principal is constrained to be the same. Choice D is wrong because the principal amount is determined by the constraint, not by order of selection.
5. A financial advisor claims that with simple interest, doubling both the principal and the time will always quadruple the interest earned, regardless of the interest rate. Which of the following best evaluates this claim?
   Answer: The claim is correct because I = Prt, so doubling P and t multiplies I by 2 × 2 = 4 When you encounter questions about simple interest claims, you need to test the mathematical relationship using the fundamental formula: $I = Prt$, where I is interest, P is principal, r is interest rate, and t is time. Let's examine what happens when you double both the principal and time. If your original interest is $I = Prt$, then with doubled principal (2P) and doubled time (2t), the new interest becomes $I = (2P) \cdot r \cdot (2t) = 4Prt$. This shows the interest is indeed quadrupled (multiplied by 4), regardless of what the interest rate actually is. The advisor's claim is mathematically sound. Looking at the incorrect options: Choice A suggests there's some maximum limit that prevents quadrupling, but simple interest has no such built-in ceiling—it grows linearly without restriction. Choice B incorrectly assumes the interest rate changes when time changes, but in simple interest problems, the rate typically remains constant throughout the investment period. Choice C creates an arbitrary condition requiring exactly 4% interest rate, but our mathematical proof shows the quadrupling effect works at any interest rate. Choice D correctly identifies that doubling both P and t in the formula $I = Prt$ results in multiplication by $2 \times 2 = 4$. Study tip: When evaluating mathematical claims about formulas, substitute the proposed changes directly into the original equation. This algebraic approach will quickly reveal whether the claim holds true and help you avoid answer choices that introduce unnecessary restrictions or false assumptions.
6. A student solves for the principal in a simple interest problem: "If $I = 150$, $r = 6\%$, and $t = 2.5$ years, find $P$". She writes: $P = \frac{I}{rt} = \frac{150}{6 \times 2.5} = \frac{150}{15} = 10$. She concludes the principal is $$10. What is wrong with this solution?
   Answer: She didn't convert the percentage rate to decimal form; the correct principal is $$1,000 The student correctly used the formula P = I/(rt) derived from I = Prt, but she failed to convert 6% to its decimal form 0.06. The correct calculation is: P = 150/(0.06 × 2.5) = 150/0.15 = $1,000. Her arithmetic (6 × 2.5 = 15, and 150/15 = 10) was correct, but she used r = 6 instead of r = 0.06. Choice A is wrong because P = I/(rt) is the correct rearrangement. Choice C is wrong because 6 × 2.5 does equal 15, not 12. Choice D confuses principal with total amount; the question asked for principal only, not principal plus interest.

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