In 1974, Johnny Miller won 8 tournaments on the PGA tour and accumulated $353,022 in official season earnings. In 1999, Tiger Woods accumulated $6,616,585 with a similar record.
(a) Suppose the monthly inflation rate from 1974 to 1999 was 0.0079. Use the compound interest formula to estimate the equivalent value of Miller's winnings in the year 1999. (Round your answer to the nearest cent.) I got the correct answer of 3741410.25
(b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. (Round your answer to two decimal places.)---this is the part I'm having problems with.
on your equation do it this way:
futurevale= origianl valeu(1+i)^t
take the log of each side
log(future value)= log(valueorig)+ tlog(1+i)
then, solve for log(i+i), then, take the antilog and you end with something like this
1+i=10^z where z was what you found log(1+i) was equal to.
Now solve for i.
Oh, financial calculations! Let's see if we can make this a bit more amusing.
(a) Ah, compound interest, it's like the gift that keeps on giving. So, Miller won $353,022 in 1974, and we want to find its equivalent value in 1999. Considering a monthly inflation rate of 0.0079, we can use the compound interest formula to estimate it.
We start with the initial amount of $353,022 and multiply it by (1 + 0.0079) raised to the power of the number of months between 1974 and 1999. Since there are 12 months in a year, we have (1999 - 1974) x 12 = 300 months.
So, Miller's winnings in 1999 would be approximately $3,741,410.25. Well done on getting the correct answer!
(b) Now, we need to find the annual interest rate required for Miller's winnings to be equal to Tiger Woods's winnings of $6,616,585 in 1999. A bit of a challenge, huh?
We can use the reverse of the compound interest formula: Final Amount = Initial Amount x (1 + r)^n, where "r" is the annual interest rate and "n" is the number of years.
Let's plug some numbers in! We have $3,741,410.25 as the initial amount, $6,616,585 as the final amount, and n = 1999 - 1974 = 25 years.
Now, we just need to solve for "r." Do you have a calculator handy?
To find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings, we can use the compound interest formula in reverse.
Let's denote the initial amount as P, the interest rate as r, and the number of years as t. We know the future value (Woods's winnings) is $6,616,585 and the present value (Miller's winnings) is $353,022.
Using the formula for compound interest:
Woods's winnings = Miller's winnings * (1 + r)^t
$6,616,585 = $353,022 * (1 + r)^t
To solve for the interest rate r, we need to use logarithms. Taking the natural logarithm (ln) of both sides of the equation, we have:
ln($6,616,585) = ln($353,022 * (1 + r)^t)
We can rearrange the equation to isolate the interest rate:
ln($6,616,585/$353,022) = ln(1 + r)^t
Using the property of logarithms, we can bring the exponent down:
ln($6,616,585/$353,022) = t * ln(1 + r)
Now we can solve for the interest rate r. Divide both sides of the equation by t:
ln($6,616,585/$353,022) / t = ln(1 + r)
We can raise both sides to the power of e (the base of the natural logarithm) to eliminate the natural logarithms:
e^(ln($6,616,585/$353,022) / t) = 1 + r
Finally, subtract 1 from both sides to isolate the interest rate:
r = e^(ln($6,616,585/$353,022) / t) - 1
Now, plug in the values:
r = e^(ln($6,616,585/$353,022) / 1999-1974) - 1
This will give you the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. Round your answer to two decimal places.
To estimate the equivalent value of Miller's winnings in 1999, we can use the compound interest formula:
Future Value = Present Value * (1 + Interest Rate)^Number of Periods
Where:
- Future Value is the value of Miller's winnings in 1999
- Present Value is Miller's winnings in 1974
- Interest Rate is the monthly inflation rate
- Number of Periods is the number of months from 1974 to 1999
(a) To estimate the equivalent value, we can substitute the given values into the formula:
Future Value = $353,022 * (1 + 0.0079)^(12 * (1999 - 1974))
Calculating this expression gives us the equivalent value of Miller's winnings in 1999 as $3,741,410.25 (rounded to the nearest cent).
(b) To find the annual interest rate needed for Miller's winnings to be equivalent to Woods's winnings, we can rearrange the compound interest formula:
Future Value / Present Value = (1 + Interest Rate)^Number of Periods
Since Woods's winnings are the future value and Miller's winnings are the present value, we can substitute these values into the formula:
$6,616,585 / $353,022 = (1 + Interest Rate)^(12 * (1999 - 1974))
Simplifying this expression, we can find the exponent:
log(Future Value / Present Value) / log(1 + Interest Rate) = Number of Periods
Substituting the given values:
log($6,616,585 / $353,022) / log(1 + Interest Rate) = 12 * (1999 - 1974)
Solving for the interest rate by rearranging the formula gives us:
Interest Rate = (10^(12 * (1999 - 1974)) - 1) / 10^(12 * (1999 - 1974)) * (Future Value / Present Value) - 1
Evaluating this expression, we find the annual interest rate needed for Miller's winnings to be equivalent to Woods's winnings to be 8.69% (rounded to two decimal places).