exercise 5 - simple & compound interest
1. explain how you can represent each time period in terms of years.
a) 4 months
b) 75 days
c) 15 weeks
2. determine the simple interest earned on each investment.
a) $450 is deposited for 4 years and earns 6.5% per year simple interest.
b) $500 is invested at 4.75% annual simple interest for 35 weeks.
c) $1100 is invested at 7.8% per year, simple interest, for 60 days.
3. determine the total number, n, of compounding periods and the interest rate, i, as a decimal, per compounding period for each scenario.
a) 6% per annum, compounded quarterly, for 3 years
b) 4.5% per annum, compounded semi - annually for 7.5 years
c) 2.4% per year, compounded monthly, for 2 years
4. avneet borrows $500 at an annual rate of 8.25% simple interest to enroll in a driver’s education course. she plans to repay the loan in 18 months.

Question

exercise 5 - simple & compound interest
1. explain how you can represent each time period in terms of years.
   a) 4 months
   b) 75 days
   c) 15 weeks
2. determine the simple interest earned on each investment.
   a) $450 is deposited for 4 years and earns 6.5% per year simple interest.
   b) $500 is invested at 4.75% annual simple interest for 35 weeks.
   c) $1100 is invested at 7.8% per year, simple interest, for 60 days.
3. determine the total number, n, of compounding periods and the interest rate, i, as a decimal, per compounding period for each scenario.
   a) 6% per annum, compounded quarterly, for 3 years
   b) 4.5% per annum, compounded semi - annually for 7.5 years
   c) 2.4% per year, compounded monthly, for 2 years
4. avneet borrows $500 at an annual rate of 8.25% simple interest to enroll in a driver’s education course. she plans to repay the loan in 18 months.

Accepted Answer

### Problem 1 (a)
# Explanation:
## Step1: Recall time conversion
To convert months to years, use the fact that 1 year = 12 months. So, divide the number of months by 12.
For 4 months: $\frac{4}{12}=\frac{1}{3}\approx0.3333$ years.
## Step2: Verify
Since each year has 12 months, the fraction of a year that 4 months represents is $\frac{4}{12}$, which simplifies to $\frac{1}{3}$ years.
# Answer: $\frac{1}{3}$ years (or approximately 0.3333 years)

### Problem 1 (b)
# Explanation:
## Step1: Recall time conversion
To convert days to years, assume a non - leap year has 365 days. So, divide the number of days by 365.
For 75 days: $\frac{75}{365}=\frac{15}{73}\approx0.2055$ years.
## Step2: Verify
Since a year is approximately 365 days, the fraction of a year that 75 days represents is $\frac{75}{365}$, which simplifies to $\frac{15}{73}$ years.
# Answer: $\frac{15}{73}$ years (or approximately 0.2055 years)

### Problem 1 (c)
# Explanation:
## Step1: Recall time conversion
To convert weeks to years, assume a year has 52 weeks. So, divide the number of weeks by 52.
For 15 weeks: $\frac{15}{52}\approx0.2885$ years.
## Step2: Verify
Since a year is approximately 52 weeks, the fraction of a year that 15 weeks represents is $\frac{15}{52}$ years.
# Answer: $\frac{15}{52}$ years (or approximately 0.2885 years)

### Problem 2 (a)
# Explanation:
## Step1: Recall simple interest formula
The simple interest formula is $I = P	imes r	imes t$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the time in years.
Given $P=\$450$, $r = 6.5\%=0.065$, and $t = 4$ years.
## Step2: Calculate interest
Substitute the values into the formula: $I=450	imes0.065	imes4$.
First, calculate $450	imes0.065 = 29.25$. Then, $29.25	imes4=117$.
# Answer: $\$117$

### Problem 2 (b)
# Explanation:
## Step1: Recall time conversion for weeks to years
First, convert 35 weeks to years. Since there are 52 weeks in a year, $t=\frac{35}{52}$ years.
The simple interest formula is $I = P	imes r	imes t$, where $P = \$500$, $r=4.75\% = 0.0475$.
## Step2: Calculate interest
Substitute the values: $I = 500	imes0.0475	imes\frac{35}{52}$.
First, calculate $500	imes0.0475 = 23.75$. Then, $23.75	imes\frac{35}{52}=\frac{23.75	imes35}{52}=\frac{831.25}{52}\approx15.9856$.
# Answer: Approximately $\$15.99$ (or more precisely $\frac{831.25}{52}\approx15.98$)

### Problem 2 (c)
# Explanation:
## Step1: Recall time conversion for days to years
Convert 60 days to years. Assuming 365 days in a year, $t=\frac{60}{365}=\frac{12}{73}$ years.
The simple interest formula is $I = P	imes r	imes t$, where $P=\$1100$, $r = 7.8\%=0.078$.
## Step2: Calculate interest
Substitute the values: $I=1100	imes0.078	imes\frac{12}{73}$.
First, calculate $1100	imes0.078 = 85.8$. Then, $85.8	imes\frac{12}{73}=\frac{85.8	imes12}{73}=\frac{1029.6}{73}\approx14.1041$.
# Answer: Approximately $\$14.10$

### Problem 3 (a)
# Explanation:
## Step1: Recall compounding period formula
The number of compounding periods $n$ is given by $n = m	imes t$, where $m$ is the number of compounding periods per year and $t$ is the number of years.
For 6% per annum, compounded quarterly: $m = 4$ (quarterly means 4 times a year), $t = 3$ years.
## Step2: Calculate $n$ and $i$
Calculate $n$: $n=4	imes3 = 12$.
The interest rate per compounding period $i=\frac{r}{m}$, where $r = 6\%=0.06$. So, $i=\frac{0.06}{4}=0.015$.
# Answer: $n = 12$, $i=0.015$

### Problem 3 (b)
# Explanation:
## Step1: Recall compounding period formula
For 4.5% per annum, compounded semi - annually: $m = 2$ (semi - annually means 2 times a year), $t = 7.5$ years.
## Step2: Calculate $n$ and $i$
Calculate $n$: $n=2	imes7.5 = 15$.
The interest rate per compounding period $i=\frac{r}{m}$, where $r = 4.5\%=0.045$. So, $i=\frac{0.045}{2}=0.0225$.
# Answer: $n = 15$, $i = 0.0225$

### Problem 3 (c)
# Explanation:
## Step1: Recall compounding period formula
For 2.4% per annum, compounded monthly: $m = 12$ (monthly means 12 times a year), $t = 2$ years.
## Step2: Calculate $n$ and $i$
Calculate $n$: $n=12	imes2=24$.
The interest rate per compounding period $i=\frac{r}{m}$, where $r = 2.4\%=0.024$. So, $i=\frac{0.024}{12}=0.002$.
# Answer: $n = 24$, $i=0.002$

### Problem 4
# Explanation:
## Step1: Recall simple interest formula for total amount
The simple interest formula for the total amount to be repaid $A=P+I$, and $I = P	imes r	imes t$.
Given $P=\$500$, $r = 8.25\%=0.0825$, and $t = 18$ months. First, convert 18 months to years: $t=\frac{18}{12}=1.5$ years.
## Step2: Calculate interest and total amount
First, calculate $I=500	imes0.0825	imes1.5$.
$500	imes0.0825 = 41.25$. Then, $41.25	imes1.5 = 61.875$.
Then, $A=P + I=500+61.875 = 561.875$.
# Answer: $\$561.88$ (rounded to the nearest cent) or $\$561.875$

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QUESTION IMAGE

Question

Explanation:

Problem 1 (a)

Step1: Recall time conversion

Step2: Verify

Step1: Recall time conversion

Step2: Verify

Step1: Recall time conversion

Step2: Verify

Answer:

Problem 1 (b)