Math / Algebra

QuestionYou want to purchase a new car in 5 years and expect the car to cost $\$ 69,000$ . Your bank offers a plan with a guaranteed APR of $6.5 \%$ if you make regular monthly deposits. How much should you deposit each month to end up with $\$ 69,000$ in 5 years?
You should invest \\square$ each month. (Do not round until the final answer. Then round to two decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to figure out how much money to put away each month to have enough to buy a $\$69,000$ car in 5 years, assuming our savings grow at a steady 6.5% annual rate. Watch out! The interest rate is annual, but we're making monthly deposits, so we need to be careful with how we use the rate!

STEP 2

1. Calculate the monthly interest rate.
2. Determine the number of periods.
3. Calculate the monthly deposit.

STEP 3

We're given an annual interest rate of 6.5%, which we write as 0.065 in decimal form.
Since we're making monthly deposits, we need to find the monthly interest rate.

STEP 4

To get the monthly rate, we divide the annual rate by 12, the number of months in a year: $\frac{0.065}{12} \approx \mathbf{0.00541667}$ So, our monthly interest rate is approximately 0.00541667.

STEP 5

We're saving for 5 years, and there are 12 months in a year.

STEP 6

So, the total number of deposit periods is $5 \cdot 12 = \mathbf{60}$ months.

STEP 7

We can use the future value of an ordinary annuity formula to figure out the monthly deposit.
The formula is: $FV = P \cdot \frac{((1 + r)^n - 1)}{r}$ Where: $FV$ is the future value (the amount we want to have at the end), $P$ is the periodic payment (the monthly deposit we're trying to find), $r$ is the periodic interest rate (the monthly rate we calculated), and $n$ is the number of periods (the total number of months).

STEP 8

We know that $FV = \$69,000$ , $r \approx 0.00541667$ , and $n = 60$ .
We want to find $P$ .
Let's plug in the values: $69000 = P \cdot \frac{((1 + 0.00541667)^{60} - 1)}{0.00541667}$

STEP 9

Let's simplify the expression inside the parentheses: $(1 + 0.00541667)^{60} \approx 1.3838$ So, the equation becomes: $69000 = P \cdot \frac{(1.3838 - 1)}{0.00541667}$

STEP 10

Now, we can simplify the fraction: $\frac{0.3838}{0.00541667} \approx 70.85$ So, the equation is now: $69000 = P \cdot 70.85$

STEP 11

To find $P$ , we divide both sides of the equation by 70.85: $P = \frac{69000}{70.85}$ $P \approx \mathbf{973.62}$

STEP 12

You should invest approximately $\$$ 973.62 each month.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the monthly interest rate.
2. Determine the number of periods.
3. Calculate the monthly deposit.

STEP 3

We're given an annual interest rate of 6.5%, which we write as 0.065 in decimal form.
Since we're making monthly deposits, we need to find the monthly interest rate.

STEP 4

To get the monthly rate, we divide the annual rate by 12, the number of months in a year: $\frac{0.065}{12} \approx \mathbf{0.00541667}$ So, our monthly interest rate is approximately 0.00541667.

STEP 5

We're saving for 5 years, and there are 12 months in a year.

STEP 6

So, the total number of deposit periods is $5 \cdot 12 = \mathbf{60}$ months.

STEP 7

STEP 8

We know that $FV = \$69,000$ , $r \approx 0.00541667$ , and $n = 60$ .
We want to find $P$ .
Let's plug in the values: $69000 = P \cdot \frac{((1 + 0.00541667)^{60} - 1)}{0.00541667}$

STEP 9

Let's simplify the expression inside the parentheses: $(1 + 0.00541667)^{60} \approx 1.3838$ So, the equation becomes: $69000 = P \cdot \frac{(1.3838 - 1)}{0.00541667}$

STEP 10

Now, we can simplify the fraction: $\frac{0.3838}{0.00541667} \approx 70.85$ So, the equation is now: $69000 = P \cdot 70.85$

STEP 11

To find $P$ , we divide both sides of the equation by 70.85: $P = \frac{69000}{70.85}$ $P \approx \mathbf{973.62}$

STEP 12

You should invest approximately $\$$ 973.62 each month.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the monthly interest rate.2. Determine the number of periods.3. Calculate the monthly deposit.

STEP 3

We're given an *annual* interest rate of **6.5%**, which we write as **0.065** in decimal form.Since we're making monthly deposits, we need to find the *monthly* interest rate.

STEP 4

To get the monthly rate, we divide the annual rate by **12**, the number of months in a year: 0.06512≈0.00541667 \frac{0.065}{12} \approx \mathbf{0.00541667} 120.065​≈0.00541667 So, our monthly interest rate is approximately **0.00541667**.

STEP 5

We're saving for **5** years, and there are **12** months in a year.

STEP 6

So, the total number of deposit periods is 5⋅12=605 \cdot 12 = \mathbf{60}5⋅12=60 months.

STEP 7

STEP 8

STEP 9

Let's simplify the expression inside the parentheses: (1+0.00541667)60≈1.3838 (1 + 0.00541667)^{60} \approx 1.3838 (1+0.00541667)60≈1.3838 So, the equation becomes: 69000=P⋅(1.3838−1)0.00541667 69000 = P \cdot \frac{(1.3838 - 1)}{0.00541667} 69000=P⋅0.00541667(1.3838−1)​

STEP 10

Now, we can simplify the fraction: 0.38380.00541667≈70.85 \frac{0.3838}{0.00541667} \approx 70.85 0.005416670.3838​≈70.85 So, the equation is now: 69000=P⋅70.85 69000 = P \cdot 70.85 69000=P⋅70.85

STEP 11

To find PPP, we divide both sides of the equation by **70.85**: P=6900070.85 P = \frac{69000}{70.85} P=70.8569000​ P≈973.62 P \approx \mathbf{973.62} P≈973.62

STEP 12

You should invest approximately $\$$ **973.62** each month.

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Studdy solves anything!

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1. Calculate the monthly interest rate.
2. Determine the number of periods.
3. Calculate the monthly deposit.

We're given an annual interest rate of 6.5%, which we write as 0.065 in decimal form.
Since we're making monthly deposits, we need to find the monthly interest rate.

To get the monthly rate, we divide the annual rate by 12, the number of months in a year: $\frac{0.065}{12} \approx \mathbf{0.00541667}$ So, our monthly interest rate is approximately 0.00541667.

We're saving for 5 years, and there are 12 months in a year.

So, the total number of deposit periods is $5 \cdot 12 = \mathbf{60}$ months.

Let's simplify the expression inside the parentheses: $(1 + 0.00541667)^{60} \approx 1.3838$ So, the equation becomes: $69000 = P \cdot \frac{(1.3838 - 1)}{0.00541667}$

Now, we can simplify the fraction: $\frac{0.3838}{0.00541667} \approx 70.85$ So, the equation is now: $69000 = P \cdot 70.85$

To find $P$ , we divide both sides of the equation by 70.85: $P = \frac{69000}{70.85}$ $P \approx \mathbf{973.62}$

You should invest approximately $\$$ 973.62 each month.