Calculating Time for US Debt to Double: Y=1/r ln(x)

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In summary, the formula for finding the number of years it takes for a loan to double in value with compound interest is Y=1/r ln(x), where r is the annual interest rate and x is the factor by which the amount of money owed increases. With an average interest rate of 5%, it would take approximately 14 years for the U.S. national debt to double if the government does not borrow or pay back any money, calculated using the natural logarithm function.
  • #1
sfgradv
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Like any loan, the government accrues interest that compounds over time on the amount it owes. If the annual interest rate is r, then the number of years it takes for the amount of money owed to increase by a factor of x is

Y=1/r ln(x)

where ln is the natural logarithm.

The average interest rate on the U.S. national debt is about 5%. If the government neither borrows any more money, nor pays back any of the money it owes, how many years will it take for the total debt to double?

Give your answer using the built-in function ln.
 
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  • #2
The idea is that you show what you already know,
or how you're thinking, so we know where you're stuck.
Since you already have the formula AND a hint,
maybe you don't know that "double" means "2".
 
  • #3
sfgradv said:
Like any loan, the government accrues interest that compounds over time on the amount it owes. If the annual interest rate is r, then the number of years it takes for the amount of money owed to increase by a factor of x is

Y=1/r ln(x)

where ln is the natural logarithm.

The average interest rate on the U.S. national debt is about 5%. If the government neither borrows any more money, nor pays back any of the money it owes, how many years will it take for the total debt to double?

Give your answer using the built-in function ln.

okay, look - it's as easy as filling in the variables with the data that's given.

Y -- that's what your trying to find
r -- that's the interest rate
x -- that's the factor by which it will change

so

r -- what is the interest rate given?
x -- what is the factor given?

it's pretty straight forward
 
  • #4
Start amount X e or 2.71 yada yada to the exponent of rate X periods
 

Related to Calculating Time for US Debt to Double: Y=1/r ln(x)

1. What does the formula Y=1/r ln(x) represent?

The formula Y=1/r ln(x) represents the amount of time it would take for the US national debt to double, where Y is the time in years, r is the average annual growth rate of the debt, and x is the starting amount of the debt.

2. How is the average annual growth rate of the debt (r) determined?

The average annual growth rate of the debt is calculated by dividing the change in the debt over a period of time by the starting amount of the debt, and then multiplying by 100 to express it as a percentage.

3. Can this formula accurately predict when the US national debt will double?

Yes, this formula can accurately predict when the US national debt will double, assuming that the average annual growth rate (r) remains constant and no major changes occur in the economy or government spending.

4. Does this formula take into account factors such as inflation and interest rates?

No, this formula does not specifically account for inflation and interest rates. However, the average annual growth rate (r) may indirectly reflect changes in these factors.

5. How can this formula be used in financial planning and decision-making?

This formula can be used as a rough estimate for predicting the future growth of the US national debt. It can also be used to compare different scenarios and make informed decisions regarding government spending and budgeting.

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