You currently have $500, and furniture you want costs $600. If you can earn 6% , how many years will you have to wait if you don't add any additional money?

Question

You currently have $500, and furniture you want costs $600. If you can earn 6% , how many years will you have to wait if you don't add any additional money?

BlackTom AI · Accepted Answer

Use the compound-interest model FV = PV(1 + r)^t because you start with a lump sum, earn 6% interest, and do not add more money. Here PV = $500, FV = $600, and r = 0.06 (6%).

Step-by-step solution:
1. Set up the equation: 600 = 500(1 + 0.06)^t.
2. Simplify the ratio: 600/500 = 1.2, so 1.2 = (1.06)^t.
3. Take natural logarithms of both sides to solve for t:
   ln(1.2) = ln((1.06)^t) = t · ln(1.06).
4. Solve for t:
   t = ln(1.2) / ln(1.06).

Intermediate numerical steps:
- ln(1.2) ≈ 0.1823215568
- ln(1.06) ≈ 0.0582689081
- t ≈ 0.1823215568 / 0.0582689081 ≈ 3.1284

Rounding to two decimal places gives t ≈ 3.13 years.

Why this is correct:
- Compound interest is the appropriate model when interest is reinvested and there are no additional contributions. The logarithm step is necessary because t is an exponent.
- If someone instead used simple interest, the formula would be FV = PV(1 + rt), yielding t = (600 − 500) / (500·0.06) = 100/30 ≈ 3.333… years, which is different because simple interest does not compound; that interpretation would be incorrect here unless the problem explicitly said "simple interest."
- If interest were compounded continuously, t = ln(1.2)/0.06 ≈ 3.04 years; again different because compounding frequency matters. The typical interpretation (annual compound) matches the calculation above.

Conclusion: Using standard annual compound interest, the time required is approximately 3.13 years, which matches the provided answer.

You currently have $500, and furniture you want costs $600. If you can earn 6% , how many years will you have to wait if you don't add any additional money? 数值题

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