Ques 1) A person thinks that in 15 years, it will cost $214,000 to provide his child with a 4-year
college education. How much more/less will the person have if he invests $75,000 today for the
next 15 years at 4 percent?
Answer1)
Step 1:
To determine how much more or less the person will have, one need to calculate how much $75,000
will grow to over the next 15 years with an interest rate of 4%. One can use the formula for
compound interest:
r nt
A=P(1+ )
n
Where:
A is the amount of money accumulated after ttt years, including interest.
P is the principal amount (the initial investment).
r is the annual interest rate (decimal form).
n is the number of times that interest is compounded per year (for simplicity, one'll assume it
compounds annually, so n=1)
t is the time the money is invested for in years.
Explanation:
The amount the investment will grow to in the future based on a given interest rate over a certain
period.
Step2:
The formula:
r nt
A=P(1+ )
n
Given:
P=75,000
r=0.04
t=15t
n=1 (compounded annually)
Now, let's calculate the future value of the investment:
, ( )
1∗15
0.04
A=75000 1+
1
A=75000 (1.04 )1∗15
A=75000∗1.8009
A=135 068
So, after 15 years, the person will have approximately $135,068.
To determine how much more or less this is compared to the cost of the child's college education
($214,000), we subtract: 214000-135068 =78932
Explanation:
It is also connected to financial planning, where the person is trying to estimate how much money
will be needed in the future to cover a major expense, such as the cost of a 4-year college education
for their child.
Final Answer:
The person will have $78,932 less than the cost of the child's 4-year college education.
, Ques2) You invested $1,200 three years ago. During the three years, you earned annual rates of
return of 4.8%, 9.2%, and 11.6%. What is the value of this investment today?
Answer2:
Step1:
To calculate the value of an investment after earning different rates of return each year, we use the
formula for compound interest for each year, adjusting for the different rates. Here's how it works:
The formula for compound interest is:
A=P×(1+r 1 )×(1+r 2 )×(1+r 3)
Where:
A is the amount of money accumulated after the given years, including interest.
P is the principal amount (initial investment).
r_1, r_2, r_3 are the annual rates of return for each year, expressed as decimals.
Explanation:
This concept involves earning interest not only on the initial investment but also on the interest that
has been accumulated over time. In this case, each year's rate of return is applied to the new value
of the investment, which includes both the principal and the interest earned in previous years.
Step2:
The formula for compound interest is
A=P×(1+r 1 )×(1+r 2 )×(1+r 3)
Given:
P=1,200
r1=4.8%=0.048
r2=9.2%=0.092
r3=11.6%=0.116
Now, we can calculate the value of the investment after 3 years:
A=1200∗( 1+0 .048 )∗( 1+0.092 )∗(1+0.116)
A=1200∗1. 048∗1.092∗1.116
A=1200∗1.277
A=1532.40
Explanation:
college education. How much more/less will the person have if he invests $75,000 today for the
next 15 years at 4 percent?
Answer1)
Step 1:
To determine how much more or less the person will have, one need to calculate how much $75,000
will grow to over the next 15 years with an interest rate of 4%. One can use the formula for
compound interest:
r nt
A=P(1+ )
n
Where:
A is the amount of money accumulated after ttt years, including interest.
P is the principal amount (the initial investment).
r is the annual interest rate (decimal form).
n is the number of times that interest is compounded per year (for simplicity, one'll assume it
compounds annually, so n=1)
t is the time the money is invested for in years.
Explanation:
The amount the investment will grow to in the future based on a given interest rate over a certain
period.
Step2:
The formula:
r nt
A=P(1+ )
n
Given:
P=75,000
r=0.04
t=15t
n=1 (compounded annually)
Now, let's calculate the future value of the investment:
, ( )
1∗15
0.04
A=75000 1+
1
A=75000 (1.04 )1∗15
A=75000∗1.8009
A=135 068
So, after 15 years, the person will have approximately $135,068.
To determine how much more or less this is compared to the cost of the child's college education
($214,000), we subtract: 214000-135068 =78932
Explanation:
It is also connected to financial planning, where the person is trying to estimate how much money
will be needed in the future to cover a major expense, such as the cost of a 4-year college education
for their child.
Final Answer:
The person will have $78,932 less than the cost of the child's 4-year college education.
, Ques2) You invested $1,200 three years ago. During the three years, you earned annual rates of
return of 4.8%, 9.2%, and 11.6%. What is the value of this investment today?
Answer2:
Step1:
To calculate the value of an investment after earning different rates of return each year, we use the
formula for compound interest for each year, adjusting for the different rates. Here's how it works:
The formula for compound interest is:
A=P×(1+r 1 )×(1+r 2 )×(1+r 3)
Where:
A is the amount of money accumulated after the given years, including interest.
P is the principal amount (initial investment).
r_1, r_2, r_3 are the annual rates of return for each year, expressed as decimals.
Explanation:
This concept involves earning interest not only on the initial investment but also on the interest that
has been accumulated over time. In this case, each year's rate of return is applied to the new value
of the investment, which includes both the principal and the interest earned in previous years.
Step2:
The formula for compound interest is
A=P×(1+r 1 )×(1+r 2 )×(1+r 3)
Given:
P=1,200
r1=4.8%=0.048
r2=9.2%=0.092
r3=11.6%=0.116
Now, we can calculate the value of the investment after 3 years:
A=1200∗( 1+0 .048 )∗( 1+0.092 )∗(1+0.116)
A=1200∗1. 048∗1.092∗1.116
A=1200∗1.277
A=1532.40
Explanation:
