Respuesta :
Answer: It takes 0.0189 years or 6.9340 days for $56 billion invested at an effective rate of 7.4% to earn an interest of $76 million.
We arrive at the answer as follows
We have
Bill Gates’ estimated net worth in 2009 = $56 billion or 56,000 million
Effective Rate of interest earned = 7.4%
Interest earned = $76 million
Let Bill Gates’ estimated net worth in 2009 be Present Value or PV
Let Bill Gates’ income after earning an interest of $76 million be Future Value or FV
Now, FV is nothing but principal invested and the interest earned during the period of investment.
So,
[tex]FV = 56,00,00,00,000 + 7,60,00,000 = 56,07,60,00,000 or 56,076 million[/tex]
Now we take a look at the FV formula with respect to time value of money. In this case,
[tex]FV = PV *(1+r)^{n}[/tex]
Substituting the values we have in the formula above, we get,
[tex]56,076 = 56,000* (1+0.074)^{n}[/tex]
[tex]\frac{56076}{56000} = 1.074^{n}[/tex]
[tex]1.001357143 = 1.074^{n}[/tex] -------- (1)
At this stage we can use log to the base 10 in order to find n.
When we convert an exponent into a log, it becomes exponent times that number. For. eg. [tex]2^{3}[/tex] becomes [tex]3 * (log 2)[/tex].
So, we can rewrite (1) above as
[tex]log_{10} (1.001357143) = {log_{10}(1.074)} * n[/tex]
Substituting the log₁₀ values in the equation above we get
[tex]0.000589 = 0.031004281n[/tex]
[tex]n = \frac{0.000589}{0.031004281} = 0.018997378 years[/tex]
Since n is less than one year, we can express it in days by multiplying the answer above with 365.
We get,
0.018997378 * 365 = 6.934043124 days
