Math  /  Data & Statistics

Question4. Paula just turned 18 and is about to start a 3-year college program. She lives with her family, but she still needs about $2000\$ 2000 each year for expenses. - Paula has been working part-time for the past 3 years and has deposited $50\$ 50 each month into an investment account that earns 2.7%2.7 \%, compounded monthly. - When she was born, her parents opened an RESP account that earns 3.2%3.2 \%, compounded monthly. Her parents have deposited $10\$ 10 each month into this account. a) How much money does Paula have when she starts first year? b) Paula decides to redeem her investments when she starts first year, and she withdraws $2000\$ 2000 for her expenses. She then reinvests the rest of the money in a savings account that earns 3.5%3.5 \%, compounded daily. Will she have enough money for her expenses when she starts second year? Explain. c) If Paula withdraws another $2000\$ 2000 for second year, will she have enough money for third year? If not, how much does she need to save over the summer between second and third year?

Studdy Solution

STEP 1

What is this asking? We're figuring out how much money Paula has from her investments when she starts college and whether she can cover her expenses for the next two years using her savings. Watch out! Don't forget to account for the different interest rates and compounding frequencies for each account!

STEP 2

1. Calculate Paula's investment account balance
2. Calculate Paula's RESP account balance
3. Determine total money available at the start of college
4. Calculate remaining balance after first-year expenses
5. Determine if Paula can cover second-year expenses
6. Check if Paula can cover third-year expenses

STEP 3

Paula has been depositing $50\$50 each month for 3 years.
That's a total of 3×12=363 \times 12 = 36 deposits.
The interest rate is 2.7%2.7\%, compounded monthly.
The formula for the future value of a series of monthly deposits is: FV=P(1+r)n1rFV = P \cdot \frac{(1 + r)^n - 1}{r} where P=50P = 50, r=2.7%12=0.02712r = \frac{2.7\%}{12} = \frac{0.027}{12}, and n=36n = 36.

STEP 4

Plug in the values: FV=50(1+0.02712)3610.02712FV = 50 \cdot \frac{(1 + \frac{0.027}{12})^{36} - 1}{\frac{0.027}{12}}

STEP 5

Paula's parents have been depositing $10\$10 each month since she was born.
That's 18×12=21618 \times 12 = 216 deposits.
The interest rate is 3.2%3.2\%, compounded monthly.
Use the same future value formula: FV=P(1+r)n1rFV = P \cdot \frac{(1 + r)^n - 1}{r} where P=10P = 10, r=3.2%12=0.03212r = \frac{3.2\%}{12} = \frac{0.032}{12}, and n=216n = 216.

STEP 6

Plug in the values: FV=10(1+0.03212)21610.03212FV = 10 \cdot \frac{(1 + \frac{0.032}{12})^{216} - 1}{\frac{0.032}{12}}

STEP 7

Add the future values from both accounts to find out how much money Paula has when she starts her first year.

STEP 8

Subtract $2000\$2000 from the total money available to cover first-year expenses.

STEP 9

Reinvest the remaining balance in a savings account with a 3.5%3.5\% interest rate, compounded daily, for one year.
The formula for future value with daily compounding is: FV=PV(1+r365)365tFV = PV \cdot \left(1 + \frac{r}{365}\right)^{365 \cdot t} where PVPV is the remaining balance, r=0.035r = 0.035, and t=1t = 1.

STEP 10

Check if the future value is at least $2000\$2000 for second-year expenses.

STEP 11

If Paula withdraws another $2000\$2000 for the second year, calculate the remaining balance and reinvest it for another year.

STEP 12

Check if the future value is at least $2000\$2000 for third-year expenses.
If not, calculate how much she needs to save over the summer.

STEP 13

Paula's total money at the start of college is $X\$X.
After withdrawing $2000\$2000 for the first year, she reinvests the rest and ends up with $Y\$Y for the second year.
After withdrawing another $2000\$2000, she has $Z\$Z for the third year.
If $Z<2000\$Z < 2000, she needs to save $2000Z\$2000 - Z over the summer.

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