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Mar 23,2025

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By investing $30 in 1850 at 6% interest compounded monthly, the investment would be worth approximately $432,390 in 2010.

Solución

Let dollars be the principal, (6% annual interest), (monthly compounding), and years (since ).
  1. The compound interest formula is
  1. Substituting the values, we have
  1. To approximate , note that
Since
we have
Thus,
  1. Multiplying by 30, we obtain
So, if the lady had invested $30 in 1850 at 6% interest compounded monthly, by 2010 the investment would be worth approximately $432,390 (rounded to the nearest dollar).

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Extra Insights

To calculate the investment compounded monthly, we can use the formula for compound interest:
Where:
  • is the amount of money accumulated after n years, including interest.
  • is the principal amount (the initial amount of money).
  • is the annual interest rate (decimal).
  • is the number of times that interest is compounded per year.
  • is the number of years the money is invested for.
In this case, , , (monthly), and (from 1850 to 2010).
Plugging this into the formula:
Solving it:
So, the investment would be worth approximately $160 in 2010, rounding to the nearest dollar.
Now, for the continuous compounding, we use the formula:
Where is the base of the natural logarithm.
Using the same , , and :
Calculating the exponent:
So:
Thus, the investment would be worth approximately $446,657 in 2010, rounding to the nearest dollar.

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