Sara wants to have $600,000 in her savings account when she retires. How much must she put in the account now, if the account pays a fixed interest rate of 8%, to ensure that she has $600,000 in 20 years?

Question

BlackTom AI · Accepted Answer

Present value (PV) equals future value divided by (1 + r)^n, so PV = 600,000 / (1.08)^20.

1. Relevant knowledge points and concepts
- Time value of money: a dollar today is worth more than a dollar in the future because of interest.  
- Present value formula for a single future sum: PV = FV / (1 + r)^n, where r is the periodic interest rate and n is the number of periods.

2. Step-by-step reasoning
- FV = $600,000, r = 8% = 0.08, n = 20 years.  
- Apply the present value formula: PV = 600,000 / (1.08)^20.

3. Intermediate calculations
- Compute powers of 1.08 up to 10 and then square to get the 20th power:
  1.08^2 = 1.1664  
  1.08^3 = 1.259712  
  1.08^4 = 1.36048896  
  1.08^5 = 1.4693280768  
  1.08^6 = 1.586874323744  
  1.08^7 = 1.71382426964352  
  1.08^8 = 1.8509302112150016  
  1.08^9 = 1.9990046281122018  
  1.08^10 = 2.158924998361178
- Square 1.08^10 to get 1.08^20:
  1.08^20 ≈ (2.158924998361178)^2 ≈ 4.661449...
- Now compute PV:
  PV = 600,000 / 4.661449... ≈ 128,728.8

4. Why the correct answer is correct
- Rounding to the nearest dollar gives PV ≈ $128,729. Investing $128,729 today at 8% compounded annually for 20 years grows to about $128,729 × 4.661449 ≈ $600,000, satisfying the requirement.

5. Why the other options are incorrect
- $231,712 would grow to roughly $231,712 × 4.661 ≈ $1,080,000, far exceeding the target (this is too large by a factor ≈1.8).  
- $180,221 grows to about $180,221 × 4.661 ≈ $840,000, also too large.  
- $139,541 grows to about $139,541 × 4.661 ≈ $650,000, still larger than $600,000.  
Each of these options is the result of incorrect application of the time-value formula (or using the wrong exponent or rate).

6. Conclusion
The correct present value is $128,729, which matches the provided answer.

Sara wants to have $600,000 in her savings account when she retires. How much must she put in the account now, if the account pays a fixed interest rate of 8%, to ensure that she has $600,000 in 20 years?单项选择题

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