A financial adviser was showing a client the value, rounded to the nearest cent, of an initial investment of $100.00...
GMAT Two Part Analysis : (TPA) Questions
A financial adviser was showing a client the value, rounded to the nearest cent, of an initial investment of \(\$100.00\) after 5, 10, 15, 20, 25, and 30 years, under the assumption that the value increases by \(\mathrm{r}\%\) per year for some positive constant r. The adviser correctly gave \(\$130.01\) for the value after 5 years. However, the adviser inadvertently made two transcription errors, and as a result, two of the remaining values shown to the client were incorrect.
Assuming the options provided are the remaining five amounts shown to the client, select for First error the lesser of the two incorrectly transcribed values and select for Second error the greater of the two incorrectly transcribed values. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visual Representation - Timeline
Let's create a timeline showing the investment growth:
$100 → $130.01 → ? → ? → ? → ? → ?
Year 0 Year 5 Year 10 Year 15 Year 20 Year 25 Year 30
Key Information
- Initial investment: $100.00
- Value after 5 years: $130.01 (correct)
- Growth rate: constant r% per year
- Two of the remaining values contain transcription errors
Phase 2: Understanding the Question
Finding the Growth Rate
Since the investment grows at a constant rate r% per year:
- $100 × (1 + r)5 = $130.01
- (1 + r)5 = 1.3001
- 1 + r = 1.3001(1/5)
Using calculator: 1.3001(1/5) ≈ 1.0539
So the annual growth rate is approximately 5.39%
What We're Looking For
We need to:
- Calculate the correct values for years 10, 15, 20, 25, and 30
- Compare with the given answer choices
- Identify which two values are incorrect
- Select the lesser error as "First error" and greater error as "Second error"
Phase 3: Finding the Answer
Calculating Correct Values
Using the compound interest formula with (1 + r)5 = 1.3001:
Year 10: $100 × (1 + r)10 = $100 × (1.3001)2 = $100 × 1.69026 ≈ $169.03
Year 15: $100 × (1 + r)15 = $100 × (1.3001)3 = $100 × 2.19764 ≈ $219.76
Year 20: $100 × (1 + r)20 = $100 × (1.3001)4 = $100 × 2.85672 ≈ $285.67
Year 25: $100 × (1 + r)25 = $100 × (1.3001)5 = $100 × 3.71288 ≈ $371.29
Year 30: $100 × (1 + r)30 = $100 × (1.3001)6 = $100 × 4.82697 ≈ $482.70
Comparing with Given Choices
Let's match our calculated values with the answer choices:
| Year | Calculated | Given Choice | Status |
| 10 | $169.03 | $160.02 | × ERROR (too low) |
| 15 | $219.76 | $219.75 | ✓ Correct (within rounding) |
| 20 | $285.67 | $285.70 | ✓ Correct (within rounding) |
| 25 | $371.29 | $317.43 | × ERROR (too low) |
| 30 | $482.70 | $482.90 | ✓ Correct (within rounding) |
Identifying the Errors
The two transcription errors are:
- $160.02 (should be ~$169.03)
- $317.43 (should be ~$371.29)
Phase 4: Solution
Since we need to select the lesser error as "First error" and the greater error as "Second error":
First error (lesser value): $160.02
Second error (greater value): $317.43