Question at position 2 How long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?5 years, 3 months8 years, 1 month6 years, 9 1/2 months7 years, 2 months

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Question at position 2 How long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?5 years, 3 months8 years, 1 month6 years, 9 1/2 months7 years, 2 months

BlackTom AI · Accepted Answer

Question recap: We want to know how long it takes for $600 to grow to $900 at an annual rate of 6% with quarterly compounding. The options are:
- 5 years, 3 months
- 8 years, 1 month
- 6 years, 9 1/2 months
- 7 years, 2 months

Step-by-step analysis of each option:

Option 1: 5 years, 3 months
- With quarterly compounding, the quarterly rate is 6%/4 = 1.5%. The accumulation factor after t years is (1.015)^{4t}. If t = 5 years and 3 months = 5.25 years, the total factor would be (1.015)^{4*5.25} = (1.015)^{21}. Numerically, (1.015)^{21} is approximately 1.343, which would imply a final amount of about 600 * 1.343 ≈ 805.8. This is short of 900, so this option underestimates the time needed.

Option 2: 8 years, 1 month
- Here t = 8.083... years. The exponent is 4t ≈ 32.333. The factor (1.015)^{32.333} is roughly about 1.65 to 1.70 depending on rounding, implying a final amount around 600 * 1.65 ≈ 990. This would overshoot 900, suggesting this option is a bit too long for reaching exactly 900, and is not a precise match to 900.

Option 3: 6 years, 9 1/2 months
- This is t = 6 + 9.5/12 = 6.7917... years. The exponent is 4t ≈ 27.1667. The factor (1.015)^{27.1667} is about 1.5 (since 1.5 is the target ratio 900/600). Indeed, solving 1.015^{4t} = 1.5 gives t ≈ 6.81 years, which corresponds very closely to 6 years 9.7 months, i.e., about 6 years 9.5 months. This option is consistent with the required growth to reach 900 from 600 under the given rate.

Option 4: 7 years, 2 months
- Here t = 7.166... years. The exponent is 4t ≈ 28.6667. The accumulation factor (1.015)^{28.6667} would be larger than 1.5, yielding a final amount greater than 900 (likely closer to 600 × about 1.55–1.60 depending on rounding). Therefore, this option would overshoot the target of 900 and does not align with the exact required growth.

Additional notes on the underlying math (for clarity):
- The exact relationship is 900 = 600 × (1 + 0.06/4)^{4t}, so (1.015)^{4t} = 1.5. Taking natural logs, 4t = ln(1.5) / ln(1.015), which gives t ≈ 0.405465 / 0.059554 ≈ 6.81 years (about 6 years 9.7 months).
- This calculation shows Option 3, which corresponds to about 6 years 9.5 months, is the closest match to the mathematically correct time. The other options deviate by a noticeable amount from this computed value, either underestimating, overshooting, or being far from the precise equivalence to 1.5 growth.

In summary, analyzing the growth factor required to move from 600 to 900 under a 6% annual rate with quarterly compounding indicates a time near 6 years 9.5 months, making the option with 6 years, 9 1/2 months the one that aligns best with the calculation. The remaining options fail to match the exact growth requirement, either producing too little or too much final amount for the given period and rate.

Question at position 2 How long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?5 years, 3 months8 years, 1 month6 years, 9 1/2 months7 years, 2 months

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