Question

In march 2009, the bill and melinda gates foundation granted $76 million to aed to fund the alive and thrive initiative. in 2009, bill gates, the cofounder and former ceo of microsoft, had an estimated net worth of $56 billion. how long does it take $56 billion invested at an effective rate of 7.4% to earn $76 million?† (round your numerical answer to four decimal places.)

Answer 1

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,  

$FV = 56,00,00,00,000 + 7,60,00,000 = 56,07,60,00,000 or 56,076 million$

Now we take a look at the FV formula with respect to time value of money. In this case,

$FV = PV *(1+r)^{n}$

Substituting the values we have in the formula above, we get,

$56,076 = 56,000* (1+0.074)^{n}$

$\frac{56076}{56000} = 1.074^{n}$

$1.001357143 = 1.074^{n}$ -------- (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. $2^{3}$ becomes $3 * (log 2)$.

So, we can rewrite (1) above as

$log_{10} (1.001357143) = {log_{10}(1.074)} * n$

Substituting the log₁₀ values in the equation above we get

$0.000589 = 0.031004281n$

$n = \frac{0.000589}{0.031004281} = 0.018997378 years$

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