1. You buy a CD with $10,000 today. Interest rate is 8%, compounding monthly. How much can you get in two years? (keep the integer) 2. You buy a CD with $10,000 today. Interest ra... 1. You buy a CD with $10,000 today. Interest rate is 8%, compounding monthly. How much can you get in two years? (keep the integer) 2. You buy a CD with $10,000 today. Interest rate is 8%, compounding annually. How much can you get in two years? (keep the integer) 3. You are saving money to buy a CD with $10,000 in 6 months. Interest rate is 8%, compounding monthly. You already have $7,000. How much more do you still need for now? (keep the integer) Read more

Steps to Solve

1. Calculate the future value with monthly compounding

We'll use the compound interest formula to find the future value (FV) of the CD with monthly compounding: $FV = P(1 + \frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the number of years.

In this case, $P = $8000$, $r = 0.04$, $n = 12$, and $t = 4$. Plugging these values into the formula: $FV = 8000(1 + \frac{0.04}{12})^{(12)(4)}$ $FV = 8000(1 + 0.003333)^{48}$ $FV = 8000(1.003333)^{48}$ $FV = 8000(1.173368)$ $FV = $9386.94$ Rounding down to the nearest integer, we get $$9386$.

2. Calculate the future value with quarterly compounding

Now, we calculate the future value with quarterly compounding. The formula remains the same, but $n$ changes to 4 (quarterly): $FV = 8000(1 + \frac{0.04}{4})^{(4)(4)}$ $FV = 8000(1 + 0.01)^{16}$ $FV = 8000(1.01)^{16}$ $FV = 8000(1.172579)$ $FV = $9380.63$ Rounding down to the nearest integer, we get $$9380$.

3. Calculate the future value with annual compounding

Next, calculate the future value with annual compounding, where $n = 1$: $FV = 8000(1 + \frac{0.04}{1})^{(1)(4)}$ $FV = 8000(1 + 0.04)^{4}$ $FV = 8000(1.04)^{4}$ $FV = 8000(1.169859)$ $FV = $9358.87$ Rounding down to the nearest integer, we get $$9358$.

4. Calculate how much more money is needed to buy the CD

The goal is to have $$10000$ at the end of 4 years. To find out how much more money is needed now to reach that goal with monthly compounding, we need to determine the present value (PV) required to reach $$10000$ in 4 years with monthly compounding.

We use the present value formula: $PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$. Here, $FV = $10000$, $r = 0.04$, $n = 12$, and $t = 4$.

$PV = \frac{10000}{(1 + \frac{0.04}{12})^{(12)(4)}}$ $PV = \frac{10000}{(1.003333)^{48}}$ $PV = \frac{10000}{1.173368}$ $PV = $8522.97$

Since you currently have $$8000$, the additional amount needed is: $$8522.97 - $8000 = $522.97$ Rounding down to the nearest integer, we need an additional $$522$.