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Answer:The smallest initial annual payment ( A ) required at a 5% interest rate is approximately $5,330.88. If the interest rate drops to 4.5% after 10 years, the Browns need to increase their annual payments by approximately $3,304.28 to reach their goal of $150,000.
Step-by-step explanation: To solve this problem, we need to break it down into two parts:
1. Finding the initial annual payment (A) with a 5% interest rate for the entire 18 year.
2. Recalculating the required payment after the interest rate changes to 4.5% after 10 years.
Part 1: Initial Annual Payment with a 5% Interest Rate
The future value of an ordinary annuity can be calculated using the formula:
[ FV = A cdot frac{(1 + r)^n - 1}{r} ]
where:
- ( FV ) is the future value of the annuity,
- ( A ) is the annual payment,
- ( r ) is the annual effective interest rate,
- ( n ) is the number of payments.
Given:
- ( FV = 150,000 ) dollars,
- ( r = 0.05 ),
- ( n = 18 ).
Substituting these values into the formula, we get:
[ 150,000 = A cdot frac{(1 + 0.05)^{18} - 1}{0.05} ]
Let's solve for ( A ):
[ 150,000 = A cdot frac{(1.05)^{18} - 1}{0.05} ]
Calculating the value of ((1.05)^{18}):
[ (1.05)^{18} approx 2.4066 ]
So,
[ 150,000 = A cdot frac{2.4066 - 1}{0.05} ]
[ 150,000 = A cdot frac{1.4066}{0.05} ]
[ 150,000 = A cdot 28.132 ]
Finally,
[ A = frac{150,000}{28.132} approx 5,330.88 ]
Part 2: Adjusted Payments After Interest Rate Drops
After 10 years, the Browns have already accumulated some amount which we need to calculate. Then we will adjust the payment for the remaining 8 years at the new interest rate.
Step 1: Calculate the accumulated amount after 10 years at 5% interest rate
Using the future value formula for the first 10 years:
[ FV_{10} = A cdot frac{(1 + r)^{10} - 1}{r} ]
[ FV_{10} = 5,330.88 cdot frac{(1.05)^{10} - 1}{0.05} ]
Calculating ((1.05)^{10}):
[ (1.05)^{10} approx 1.6289 ]
So,[ FV_{10} = 5,330.88 cdot frac{1.6289 - 1}{0.05} ]
[ FV_{10} = 5,330.88 cdot frac{0.6289}{0.05} ]
[ FV_{10} = 5,330.88 cdot 12.578 approx 67,059.62 ]
Step 2: Calculate the new annual payment needed for the next 8 years at 4.5% interest rate
The remaining amount needed is:
[ 150,000 - 67,059.62 = 82,940.38 ]
Using the future value formula with the new interest rate ( r = 0.045 ) and ( n = 8 ):
[ 82,940.38 = A_{new} cdot frac{(1 + 0.045)^8 - 1}{0.045} ]
Calculating ((1.045)^8):
[ (1.045)^8 approx 1.4323 ]
So,
[ 82,940.38 = A_{new} cdot frac{1.4323 - 1}{0.045} ]
[ 82,940.38 = A_{new} cdot frac{0.4323}{0.045} ]
[ 82,940.38 = A_{new} cdot 9.6067 ]
Finally,
[ A_{new} = frac{82,940.38}{9.6067} approx 8,635.16 ]
Increase in Payments
The Browns must increase their annual payment from ( 5,330.88 ) to ( 8,635.16 ).
[ text{Increase in payment} = 8,635.16 - 5,330.88 approx 3,304.28]
Conclusion
The smallest initial annual payment ( A ) required at a 5% interest rate is approximately $5,330.88. If the interest rate drops to 4.5% after 10 years, the Browns need to increase their annual payments by approximately $3,304.28 to reach their goal of $150,000.
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