Futurama’s Investment Mystery: How Long Does it Take for a Penny to Turn into a Billion?

In the iconic episode of Futurama, “The Why of Fry,” we are presented with a fascinating scenario where an initial investment of 93 cents grows to an astounding 4.3 billion dollars over a mere 1000 years. This prompts the question: with today’s interest rates, how long would it take for the same amount of money to achieve this growth?

Understanding Compound Interest

To solve this puzzle, we can use the formula for compound interest:

A P(1 r)t

Where:

A the amount of money accumulated after n years, including interest. P the principal amount (initial investment). r the annual interest rate (decimal). t the number of years the money is invested.

Example Calculation

Let’s assume an initial investment of 1000 dollars and use an average deposit rate of about 0.5% (0.005). We want to see how long it would take for this investment to grow to 4.3 billion dollars.

Initial investment, P: 1000
Target amount, A: 4,300,000,000
Annual interest rate, r: 0.005

Plugging the values into the formula:

4,300,000,000 1000(1 0.005)t

This simplifies to:

4,300,000 (1.005)t

Next, we take the logarithm of both sides:

log(4,300,000) t log(1.005)

Now we can solve for t:

t log(4,300,000) / log(1.005)

Using a calculator:

log(4,300,000) ≈ 6.633
log(1.005) ≈ 0.00217

Substitute these values into the equation:

t ≈ 6.633 / 0.00217 ≈ 3055.8

Conclusion

To achieve a future value of 4.3 billion dollars with an initial investment of 1000 dollars at an annual interest rate of 0.5%, it would take approximately 3056 years. This illustrates the power of compound interest, even with relatively modest annual returns.

Realistic Rate of Return

David X. Cohen, one of the creators of Futurama and The Simpsons, has a background in both physics and computer science, which explains the mathematical accuracy in the show. The equation used in the show would be:

0.93(1 r)1000 4.3 times 10^9

This simplifies to:

(1 r) (4.3 times 10^9 / 0.93)^(1/1000)

r 0.0225 (2.25%)

The 2.25% annual return is more in line with the interest rates 20 years ago, when savings account yields were higher.

Modern Day Calculation

Assuming a more conservative current rate of return, say 0.75% (0.0075), how long would it take to achieve the same amount:

1.0075x 4.3 times 10^9

x ln(1.0075) ln(4.3 times 10^9)

x ln(4.3 times 10^9) / ln(1.0075)

Using a calculator:

x ≈ ln(4.3 times 10^9) / 0.0075 ≈ 2978 years

At a 1% annual rate, the amount of time would be:

(1.01)x 4.3 times 10^9

x ln(1.01) ln(4.3 times 10^9)

x ln(4.3 times 10^9) / ln(1.01)

x ≈ ln(4.3 times 10^9) / 0.01 ≈ 2237 years

Conclusion

Based on current interest rates, achieving a future value of 4.3 billion dollars from an initial investment of 93 cents would take a significantly longer time. The dichotomy between the show’s setting and modern-day economics highlights the importance of compound interest and the impact of interest rates over long periods.

Futurama and Mathematical Humor

David X. Cohen, a physics and computer science graduate, clearly infused Futurama with mathematical humor, ensuring that the audience would appreciate the depth of the characters’ scenarios, stories and puzzles. His expertise enriches the show’s storyline and adds a layer of complexity that engages viewers with both entertainment and education.