Amagat's law of additive volumes states that we can simply add up the individual volumes of each gas (provided they are at the same temperature and pressure) to get the total volume of the mixture. Conservation of volume is an acceptable assumption for gases (but not always for liquid mixtures). This works for gases since the molecules are very small and only take up a minimal amount of space in a gas.
There are simple equation you can use the compute the amount of money you will have when each of the three situations given in the questions are applied. The following illustrates for each scenario.
Simple Interest Rate Scenario
Simple interest is the interest rate that is applied only on the original principal amount. So in this scenario, the 6.6% interest is rate is the rate of interest that Katherine will continue to get on her initial deposit of $500 only even though the balance in the account will increase each year by the amount of interest rate she receives. So after 13 years, 13 years worth of interest will be applied on the original deposit.
The mathematical equation is: FV = P x (1 + RT)
Where, FV is the future value, P is the principal amount, R is the interest rate and T is the time period. When we plug in the values the equation would become:
FV = 500 x [1 + (0.066 x 13)] = $929
Annual Compound Interest Scenario
Compound interest, as opposed to simple interest, is a situation in which the interest rate is being applied on the available balance in the account. So, this balance would include the amount of interest accumulated over prior periods as well. So each year, 6.6% will be applied on the $500 PLUS the amount of interest accumulate.
The mathematical formula includes the same variables but is different in that, FV = P x (1 + R)^T.
Plugging in the values, FV = 500 x (1 + 0.066)^13 = $1,147.66
Quarterly Compounded Interest Scenario
This is similar to the annual compound interest scenario in that interest is applied on both the principal and the prior years' accumulated interest but rather than being applied annually, it will be applied quarterly.
The mathematical formula is FV = P x (1 + . The new variable added is "N" which refers to the number of times the interest is compounded which in the question's context is 4 (quarterly compounding).
So the formula becomes, FV = 500 x (1 + = $1,170.99