12. stephanie has created a study tool to help her study compound interest. she writes the compound…
12. stephanie has created a study tool to help her study compound interest. she writes the compound - interest formula with letters different than the traditional representation. $x = m(1+\frac{q}{k})^{kb}$\n a. if q is increased, does the new balance increase or decrease? explain your answer.\n b. if k is decreased, does the new balance increase or decrease? explain.\n c. if b is increased, does the new balance increase or decrease? explain.\n d. is it possible that m > x? explain.\n e. using stephanies variable representation, express the amount of interest earned on the account.\n13. compare the simple interest for one year on a principal of 1 million dollars at an interest rate of 6.3% to compounding every second for the same principal and interest rate.\n a. how many seconds are in an hour?\n b. how many seconds are in a day?\n c. how many seconds are in a year?\n d. how much interest does $1,000,000 earn in one year at 6.3% interest, compounded every second?\n e. how much does the same $1,000,000 earn at 6.3% in one year, under simple interest?\n f. how much more interest did the compounded every - second account earn when compared to the simple - interest account?\n14. britney invested $4,000 in a cd at ttyl bank that pays 1.4% interest compounded monthly.\n a. how much will britney have in her account at the end of one year?\n b. what is the apy for this account? round to the nearest hundredth of a percent.\n15. how much more would $5,000 earn in 10 years, compounded daily at 2% when compared to the interest on $5,000 over 10 years, at 2% compounded semiannually?
Answer
12.
a.
Explanation:
Step1: Analyze the formula
The compound - interest formula is $X = M(1+\frac{Q}{K})^{KB}$. When $Q$ is increased, the value of $(1 + \frac{Q}{K})$ increases. Since $M>0$, $K>0$, $B>0$, and the exponent $KB>0$, as $(1+\frac{Q}{K})$ increases, $(1+\frac{Q}{K})^{KB}$ increases, and thus $X$ (the new balance) increases.
Answer:
The new balance increases. As $Q$ increases, the value of $(1+\frac{Q}{K})$ increases, and since all other variables are non - negative, the overall value of $X$ increases.
b.
Explanation:
Step1: Analyze the formula
When $K$ is decreased, the value of $\frac{Q}{K}$ increases. So, the value of $(1+\frac{Q}{K})$ increases. Since $KB>0$ and $M > 0$, as $(1+\frac{Q}{K})$ increases, $(1+\frac{Q}{K})^{KB}$ increases, and $X$ (the new balance) increases.
Answer:
The new balance increases. Decreasing $K$ makes $\frac{Q}{K}$ larger, increasing $(1+\frac{Q}{K})$ and thus $(1+\frac{Q}{K})^{KB}$, which in turn increases $X$.
c.
Explanation:
Step1: Analyze the formula
If $B$ (the number of years) is increased, and since $(1+\frac{Q}{K})> 1$ (assuming $Q>0,K>0$), as the exponent $KB$ increases, the value of $(1+\frac{Q}{K})^{KB}$ increases. Since $M>0$, $X = M(1+\frac{Q}{K})^{KB}$ increases.
Answer:
The new balance increases. Increasing the exponent $KB$ (by increasing $B$) of a number greater than 1, $(1+\frac{Q}{K})$, will increase the value of $(1+\frac{Q}{K})^{KB}$, and thus increase $X$.
d.
Explanation:
Step1: Recall compound - interest concept
In compound interest, $X = M(1+\frac{Q}{K})^{KB}$. Since $(1+\frac{Q}{K})^{KB}>1$ for $Q > 0,K>0,B>0$, then $M(1+\frac{Q}{K})^{KB}>M$ (because $M>0$).
Answer:
It is not possible that $M > X$. In compound interest, with positive interest rate ($Q>0$), positive compounding frequency ($K>0$) and positive time ($B>0$), the final amount $X$ is always greater than the principal $M$.
e.
Explanation:
Step1: Recall interest formula
The interest earned $I$ is the final amount minus the principal. Given $X = M(1+\frac{Q}{K})^{KB}$, then $I=X - M$. So, $I = M(1+\frac{Q}{K})^{KB}-M$.
Answer:
The amount of interest earned is $M(1+\frac{Q}{K})^{KB}-M$.
13.
a.
Explanation:
Step1: Use time conversion
There are 60 seconds in a minute and 60 minutes in an hour. So, the number of seconds in an hour is $60\times60=3600$.
Answer:
3600 seconds
b.
Explanation:
Step1: Use time conversion
There are 24 hours in a day and 3600 seconds in an hour. So, the number of seconds in a day is $24\times3600 = 86400$.
Answer:
86400 seconds
c.
Explanation:
Step1: Assume a non - leap year
A non - leap year has 365 days. Since there are 86400 seconds in a day, the number of seconds in a non - leap year is $365\times86400=31536000$.
Answer:
31536000 seconds
d.
Explanation:
Step1: Identify compound - interest formula
The compound - interest formula is $A=P(1 +\frac{r}{n})^{nt}$, where $P = 1000000$, $r=0.063$, $n = 31536000$ (seconds in a year), and $t = 1$.
Step2: Calculate
$A=1000000(1+\frac{0.063}{31536000})^{31536000\times1}$. Let $x=\frac{0.063}{31536000}\approx1.9977\times10^{-9}$. Then $(1 + x)^{n}\approx e^{nx}$ (using the approximation $(1+\frac{a}{n})^{n}\approx e^{a}$ for large $n$). Here, $nx=0.063$. So, $A\approx1000000e^{0.063}\approx1000000\times1.065008\approx1065008$. The interest earned $I=A - P=1065008 - 1000000=65008$.
Answer:
Approximately $$65008$
e.
Explanation:
Step1: Use simple - interest formula
The simple - interest formula is $I=Prt$. Here, $P = 1000000$, $r=0.063$, and $t = 1$.
Step2: Calculate
$I=1000000\times0.063\times1 = 63000$.
Answer:
$$63000$
f.
Explanation:
Step1: Subtract simple interest from compound interest
The interest from compounding every second is approximately $$65008$ and the simple interest is $$63000$.
Step2: Calculate the difference
$65008-63000 = 2008$.
Answer:
$$2008$
14.
a.
Explanation:
Step1: Identify compound - interest formula
The compound - interest formula for monthly compounding is $A=P(1+\frac{r}{n})^{nt}$, where $P = 4000$, $r=0.014$, $n = 12$, and $t = 1$.
Step2: Calculate
$A=4000(1+\frac{0.014}{12})^{12\times1}$. First, $\frac{0.014}{12}\approx0.001167$. Then $(1 + 0.001167)^{12}\approx1.01409$. So, $A=4000\times1.01409 = 4056.36$.
Answer:
$$4056.36$
b.
Explanation:
Step1: Recall APY formula
The APY formula is $APY=(1+\frac{r}{n})^{n}-1$. Here, $r = 0.014$ and $n = 12$.
Step2: Calculate
$(1+\frac{0.014}{12})^{12}-1\approx1.01409 - 1=0.01409\approx1.41%$.
Answer:
$1.41%$
15.
Explanation:
Step1: For daily compounding
The compound - interest formula is $A_1=P(1+\frac{r}{n_1})^{n_1t}$, where $P = 5000$, $r=0.02$, $n_1 = 365$, and $t = 10$. $A_1=5000(1+\frac{0.02}{365})^{365\times10}$. Let $x=\frac{0.02}{365}\approx0.00005479$. Then $(1 + x)^{3650}\approx1.22139$. So, $A_1=5000\times1.22139 = 6106.95$.
Step2: For semi - annual compounding
The compound - interest formula is $A_2=P(1+\frac{r}{n_2})^{n_2t}$, where $n_2 = 2$, $P = 5000$, $r=0.02$, and $t = 10$. $A_2=5000(1+\frac{0.02}{2})^{2\times10}=5000(1 + 0.01)^{20}$. $(1.01)^{20}\approx1.22019$. So, $A_2=5000\times1.22019 = 6100.95$.
Step3: Calculate the difference
The difference in interest is $A_1 - A_2=6106.95 - 6100.95 = 6$.
Answer:
$$6$