Question 1165513: Fill in the missing vaules in the table below. Show all work in the space provided. (Determine your answer algebraically for part A and B)
Q.[Present Value] [Annual Rate] [Compounded] [Time] [Future Value] [Interest]
Earned
A. $500 4 1/2 Semi-Annually 2 years _______ ______
B. _____ 5 Quarterly 6 years $1616.83 ______
C. $1250 _______ Monthly 16 months $1381.02 ______
N= I%= PV= PMT= FV= P/Y= C/Y=
D. $875 10.5% Daily ________ $2025 _______
N= I%= PV= PMT= FV= P/Y= C/Y=
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! ```python?code_reference&code_event_index=2
import numpy as np
# --- Part A Calculations ---
PV_A = 500
r_A = 0.045
m_A = 2
t_A = 2
FV_A = PV_A * (1 + r_A / m_A)**(m_A * t_A)
I_A = FV_A - PV_A
# --- Part B Calculations ---
FV_B = 1616.83
r_B = 0.05
m_B = 4
t_B = 6
PV_B = FV_B * (1 + r_B / m_B)**(-m_B * t_B)
I_B = FV_B - PV_B
# --- Part C Calculations ---
PV_C = 1250
FV_C = 1381.02
m_C = 12
t_C_months = 16
t_C = t_C_months / 12
mt_C = m_C * t_C
r_C = m_C * ((FV_C / PV_C)**(1/mt_C) - 1)
I_C = FV_C - PV_C
# --- Part D Calculations ---
PV_D = 875
FV_D = 2025
r_D = 0.105
m_D = 365
# Time t = ln(FV/PV) / (m * ln(1 + r/m))
ratio_D = FV_D / PV_D
log_ratio_D = np.log(ratio_D)
log_term_D = m_D * np.log(1 + r_D / m_D)
t_D = log_ratio_D / log_term_D
I_D = FV_D - PV_D
N_D_periods = t_D * m_D
print(f"A: FV={FV_A:.2f}, I={I_A:.2f}")
print(f"B: PV={PV_B:.2f}, I={I_B:.2f}")
print(f"C: r={r_C:.4f} (Rate is 100 * r), I={I_C:.2f}")
print(f"D: t={t_D:.4f} years, I={I_D:.2f}, N_periods={N_D_periods:.2f}")
```
```text?code_stdout&code_event_index=2
A: FV=546.54, I=46.54
B: PV=1200.01, I=416.82
C: r=0.0750 (Rate is 100 * r), I=131.02
D: t=7.9926 years, I=1150.00, N_periods=2917.29
```
Here are the missing values and the step-by-step algebraic work for each part.
The formula used for compound interest is:
$$FV = PV \left(1 + \frac{r}{m}\right)^{mt}$$
Where $r$ is the annual rate as a decimal, $m$ is the compounding frequency, and $t$ is the time in years.
-----
## Part A: Find Future Value ($FV$) and Interest Earned ($I$)
| $\mathbf{PV}$ | $\mathbf{r}$ | $\mathbf{m}$ | $\mathbf{t}$ | $\mathbf{FV}$ | $\mathbf{I}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $\$500$ | $4.5\%$ | 2 (Semi-Annually) | 2 years | **$546.54** | **$46.54** |
### Work:
**1. Calculate Future Value ($FV$)**
Given: $PV=500$, $r=0.045$, $m=2$, $t=2$.
$$FV = 500 \left(1 + \frac{0.045}{2}\right)^{2 \times 2}$$
$$FV = 500 (1.0225)^4$$
$$FV \approx 500 \times 1.09308$$
$$\mathbf{FV \approx \$546.54}$$
**2. Calculate Interest Earned ($I$)**
$$I = FV - PV$$
$$I = 546.54 - 500$$
$$\mathbf{I = \$46.54}$$
-----
## Part B: Find Present Value ($PV$) and Interest Earned ($I$)
| $\mathbf{PV}$ | $\mathbf{r}$ | $\mathbf{m}$ | $\mathbf{t}$ | $\mathbf{FV}$ | $\mathbf{I}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| **$1200.01** | $5\%$ | 4 (Quarterly) | 6 years | $\$1616.83$ | **$416.82** |
### Work:
**1. Calculate Present Value ($PV$)**
Given: $FV=1616.83$, $r=0.05$, $m=4$, $t=6$.
$$PV = FV \left(1 + \frac{r}{m}\right)^{-mt}$$
$$PV = 1616.83 \left(1 + \frac{0.05}{4}\right)^{-4 \times 6}$$
$$PV = 1616.83 (1.0125)^{-24}$$
$$PV \approx 1616.83 \times 0.74247$$
$$\mathbf{PV \approx \$1200.01}$$
**2. Calculate Interest Earned ($I$)**
$$I = FV - PV$$
$$I = 1616.83 - 1200.01$$
$$\mathbf{I = \$416.82}$$
-----
## Part C: Find Annual Rate ($r$) and Interest Earned ($I$)
| $\mathbf{PV}$ | $\mathbf{r}$ | $\mathbf{m}$ | $\mathbf{t}$ | $\mathbf{FV}$ | $\mathbf{I}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $\$1250$ | **7.50%** | 12 (Monthly) | 16 months | $\$1381.02$ | **$131.02** |
### Work:
**1. Calculate Annual Rate ($r$)**
Given: $PV=1250$, $FV=1381.02$, $m=12$, $mt=16$ (periods).
$$r = m \left[ \left(\frac{FV}{PV}\right)^{\frac{1}{mt}} - 1 \right]$$
$$r = 12 \left[ \left(\frac{1381.02}{1250}\right)^{\frac{1}{16}} - 1 \right]$$
$$r = 12 \left[ (1.104816)^{\frac{1}{16}} - 1 \right]$$
$$r \approx 12 [1.00625 - 1]$$
$$r \approx 12 [0.00625]$$
$$r \approx 0.0750$$
$$\mathbf{r = 7.50\%}$$
**2. Calculate Interest Earned ($I$)**
$$I = FV - PV$$
$$I = 1381.02 - 1250$$
$$\mathbf{I = \$131.02}$$
**TVM Variables:**
| $\mathbf{N}$ | $\mathbf{I\%}$ | $\mathbf{PV}$ | $\mathbf{PMT}$ | $\mathbf{FV}$ | $\mathbf{P/Y}$ | $\mathbf{C/Y}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 16 | 7.50 | -1250 | 0 | 1381.02 | 12 | 12 |
-----
## Part D: Find Time ($t$) and Interest Earned ($I$)
| $\mathbf{PV}$ | $\mathbf{r}$ | $\mathbf{m}$ | $\mathbf{t}$ | $\mathbf{FV}$ | $\mathbf{I}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $\$875$ | $10.5\%$ | 365 (Daily) | **7.99 years** | $\$2025$ | **$1150.00** |
### Work:
**1. Calculate Time ($t$)**
Given: $PV=875$, $FV=2025$, $r=0.105$, $m=365$.
$$t = \frac{\ln\left(\frac{FV}{PV}\right)}{m \ln\left(1 + \frac{r}{m}\right)}$$
$$t = \frac{\ln\left(\frac{2025}{875}\right)}{365 \ln\left(1 + \frac{0.105}{365}\right)}$$
$$t = \frac{\ln(2.3142857)}{365 \ln(1.00028767)}$$
$$t \approx \frac{0.83907}{365 \times 0.00028763}$$
$$t \approx \frac{0.83907}{0.10500}$$
$$\mathbf{t \approx 7.99 \text{ years}}$$
**2. Calculate Interest Earned ($I$)**
$$I = FV - PV$$
$$I = 2025 - 875$$
$$\mathbf{I = \$1150.00}$$
**TVM Variables:**
| $\mathbf{N}$ | $\mathbf{I\%}$ | $\mathbf{PV}$ | $\mathbf{PMT}$ | $\mathbf{FV}$ | $\mathbf{P/Y}$ | $\mathbf{C/Y}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2917.29 | 10.5 | -875 | 0 | 2025 | 365 | 365 |
*(Note: $N$ is the total number of compounding periods, $N = mt \approx 365 \times 7.9926 \approx 2917.29$ days)*
|
|
|
