Joan won a multi-million dollar lottery. She
decides to give $1 000 000 of her winnings
to charity. Her plan is to give 1/2 , or 2�^-1
, to
charity in January, and then give half of the
remaining amount in February, half again in
March, and so on.
a) What fraction remains after 6 months?
Can this be put in the following formula to be solved?
=inital amount*(1/2)^(t/h)
1/2 + 1/4 + ... + 1/2^n = 1 - 1/2^n
so, after n months, 1/2^n is left
after 6 months, 1/64 is left
Yes, in this case, the formula = initial amount * (1/2)^(t/h) can be used to calculate the fraction that remains after a certain number of months.
In the given scenario, Joan plans to give half of the remaining amount to charity every month. So, the initial amount is $1,000,000 and the time period is 6 months.
We can substitute these values into the formula to find the fraction that remains after 6 months:
Fraction that remains = $1,000,000 * (1/2)^(6/1)
Simplifying the equation:
Fraction that remains = $1,000,000 * (1/2)^6
= $1,000,000 * (1/64)
= $15,625
Therefore, after 6 months, a fraction of $15,625 remains from Joan's initial amount of $1,000,000.
Yes, this situation can be represented using the formula you provided:
remaining amount = initial amount * (1/2)^(t/h)
Where:
- remaining amount represents the money left after a certain time period
- initial amount represents the total amount won in the lottery
- t represents the number of time periods (in this case, months)
- h represents the length of each time period (in this case, 1 month)
To find the fraction that remains after 6 months, we can plug in the values into the formula:
t = 6 (since we want to calculate after 6 months)
h = 1 (since the time periods are months)
Using the formula, we can calculate the fraction that remains after 6 months by substituting these values:
remaining amount = initial amount * (1/2)^(6/1)
Now we need to calculate (1/2)^(6/1):
(1/2)^(6/1) = (1/2)^6 = 1/64
So, substituting this back into the formula, we have:
remaining amount = initial amount * (1/64)
This means that after 6 months, only 1/64 of the initial amount will remain.