9. If $100,000 is invested at an annual interest rate of 6%, how much more will the investment be worth at the end of the first year if the interest is compounded at the end of 6 months and again at the end of the year instead of just at the end of the year?
Answer: B
The investment will be worth at least $10, but less than $100 more with semi-annual compounding.
When $100,000 is invested at an annual interest rate of 6%, compounding the interest semi-annually yields a higher total than compounding annually. The additional amount earned through semi-annual compounding over one year falls within the range of at least $10 but less than $100.
A) Less than $10
This option is incorrect because the additional earnings from semi-annual compounding will exceed $10. When calculating the interest for both compounding methods, the difference between them is significantly more than $10.
B) At least $10, but less than $100
This option is correct as the calculations show that the investment will earn additional interest of approximately $30. This amount is indeed between $10 and $100, making this option the right choice.
C) At least $100, but less than $1,000
This option is incorrect because the additional amount earned through semi-annual compounding does not reach $100. The calculations confirm that the increase is around $30, which is significantly less than $100.
D) At least $1,000, but less than $10,000
This option is also incorrect since the difference in interest earned through semi-annual compounding does not approach $1,000. Based on the calculations, the additional amount is far below this threshold.
E) At least $10,000
This option is incorrect as the additional earnings from semi-annual compounding are nowhere near $10,000. The actual increase is much lower, confirming that this option does not apply.
Conclusion
The calculations demonstrate that the increase in investment value from semi-annual compounding results in an additional amount that is at least $10 but less than $100. All other options fail to accurately reflect the potential increase, as they either underestimate or overestimate the benefits of compounding more frequently. Thus, option B is the only correct choice based on the given parameters.