Suppose a young couple deposits $700 at the end of each quarter in an account that earns 7.1%, compounded quarterly, for a period of 6 years. After the 6 years, they start a family and find they can contribute only $200 per quarter. If they leave the money from the first 6 years in the account and continue to contribute $200 at the end of each quarter for the next 18½ years, how much will they have in the account (to help with their child's college expenses)?
Part 1:
R=$700 per quarter
i=7.1%/4 per quarter
n=6*4=24 quarters
S=future value
=R((1+i)^n-1)/i
=700((1+.071/4)^24-1)/(.071/4)
=$20,720.40
After 6 years,
S1=future value (after 18.5 years) of initial investment
=S(1+i)^n
=$20720.40(1+.071/4)^74
=$76179.91
R=$200 per quarter
i=7.1%/4 per quarter
n=18.5*4=74 quarters
S2=future value for further contributions
=200((1+.071/4)^74-1)/(.071/4)
=$30,158.48
Total future value
= $76179.91+$30158.48
= $106,338.39
Check the calculations please.
Well, let me put on my math clown nose for this one!
For the first 6 years, let's calculate the amount they would have accumulated. The interest rate is 7.1% compounded quarterly, which means it's like getting paid every three months for being financially responsible.
The interest rate is like the fairy of compound growth sprinkling magic dust on their money. So, let's calculate how much money they would have after 6 years:
Principal = $700
Interest rate = 7.1% per year = 1.775% per quarter
Number of quarters in 6 years = 6 * 4 = 24
Now, with my super clown calculator:
Amount after 6 years = Principal * (1 + (Interest rate/100))^Number of quarters
Amount after 6 years = $700 * (1 + (1.775/100))^24
Drumroll, please...
After 6 years, they would have approximately $9,171.71 in the account.
But wait, there's more!
Now they start contributing $200 each quarter for another 18½ years (aka a really long time). Let's calculate that super accumulation of money:
Amount after 18.5 years = Amount after 6 years * (1 + (1.775/100))^74
Amount after 18.5 years = $9,171.71 * (1 + (1.775/100))^74
And the grand total is...
After 18.5 years, they would have approximately $49,621.38 in the account.
So, with a lot of saving, a little magic dust of compound growth, and a sprinkle of humor, they'll have a nice sum for their child's college expenses.
To calculate the future value of the account, we can use the formula for compound interest:
Future Value = Present Value * (1 + Interest Rate / Number of Compounding Periods)^(Number of Compounding Periods * Number of Years)
For the first 6 years, the young couple deposits $700 at the end of each quarter, and the account earns 7.1% interest compounded quarterly.
Present Value = $700 * 4 quarters per year = $2,800 (since they make quarterly deposits)
Interest Rate = 7.1% = 0.071
Number of Compounding Periods per year = 4 (quarterly interest)
Number of Years = 6
Future Value (after 6 years) = $2,800 * (1 + 0.071/4)^(4 * 6)
Future Value (after 6 years) = $2,800 * (1.01775)^(24)
Future Value (after 6 years) ≈ $3,763.47
After the first 6 years, they continue to make quarterly deposits of $200 for 18.5 years.
Quarterly Deposit = $200
Number of Compounding Periods per year = 4 (quarterly interest)
Number of Years = 18.5
Future Value (after additional 18.5 years) = $200 * ((1 + 0.071/4)^(4 * 18.5) - 1) / (0.071/4)
Future Value (after additional 18.5 years) ≈ $26,440.45
Total Future Value = Future Value (after 6 years) + Future Value (after additional 18.5 years)
Total Future Value ≈ $3,763.47 + $26,440.45
Total Future Value ≈ $30,203.92
Therefore, they will have approximately $30,203.92 in the account to help with their child's college expenses.
To calculate how much the young couple will have in the account after the specified period, we can break down the problem into two parts:
1. Calculate the amount in the account after the initial 6 years.
2. Calculate the amount in the account after the remaining 18½ years.
Let's first calculate the amount in the account after the initial 6 years.
Step 1: Calculate the interest rate per period.
The interest rate per period is given as 7.1%, compounded quarterly. To convert this annual interest rate to the interest rate per quarter, divide it by 4 (since there are 4 quarters in a year):
Interest rate per quarter = 7.1% / 4 = 1.775%.
Step 2: Calculate the number of periods.
Since the couple deposits $700 at the end of each quarter, and they do this for 6 years, which consists of 4 quarters in a year, the total number of periods will be:
Number of periods = 6 years x 4 quarters/year = 24 quarters.
Step 3: Calculate the amount in the account after the initial 6 years.
We can use the formula for the future value of an ordinary annuity to find the amount in the account after the initial 6 years:
FV = P * [(1 + r)^n - 1] / r,
where FV is the future value, P is the periodic payment, r is the interest rate per period, and n is the number of periods.
Using the provided values:
P = $700 (they deposit $700 at the end of each quarter),
r = 1.775% (interest rate per quarter),
n = 24 (number of quarters).
Plugging these values into the formula, we can calculate the future value after the initial 6 years.
Step 4: Calculate the future value after the initial 6 years.
FV = $700 * [(1 + 0.01775)^24 - 1] / 0.01775
≈ $700 * (1.01775^24 - 1) / 0.01775
≈ $700 * (1.499899585 - 1) / 0.01775
≈ $700 * (0.499899586) / 0.01775
≈ $700 * 28.1560593
≈ $19,709.24
After the initial 6 years, the amount in the account will be approximately $19,709.24.
Now let's calculate the amount in the account after the remaining 18½ years.
Step 1: Calculate the number of periods.
Since the couple will contribute $200 at the end of each quarter for the remaining 18½ years, the total number of periods will be:
Number of periods = 18½ years x 4 quarters/year = 74 quarters.
Step 2: Calculate the future value after the remaining 18½ years.
Using the same formula as before, and using the new values:
P = $200 (now they contribute $200 at the end of each quarter),
r = 1.775% (interest rate per quarter),
n = 74 (number of quarters).
FV = $200 * [(1 + 0.01775)^74 - 1] / 0.01775 (similar calculation as before)
After the remaining 18½ years, the amount in the account will be approximately equal to the calculated future value.
To find the total amount in the account, we add the amount after the initial 6 years to the amount after the remaining 18½ years:
Total amount = $19,709.24 + amount after the remaining 18½ years.
This will give us the final answer.