Math / Algebra

QuestionIf 8000 dollars is invested in a bank account at an interest rate of 7 per cent per year, Find the amount in the bank after 13 years if interest is compounded annually: $\square$ Find the amount in the bank after 13 years if interest is compounded quaterly: $\square$ Find the amount in the bank after 13 years if interest is compounded monthly: $\square$
Round all answers to 2 decimal places.

Studdy Solution

STEP 1

What is this asking? We need to figure out how much money we'll have after 13 years with different ways of compounding interest. Watch out! Don't forget to convert the percentage to a decimal!
Also, remember that compounding quarterly means 4 times a year, and monthly means 12 times a year!

STEP 2

1. Calculate annually compounded amount
2. Calculate quarterly compounded amount
3. Calculate monthly compounded amount

STEP 3

Alright, let's start with the annual compounding!
Our initial investment is $\$8000$ , and the interest rate is $7\%$ or $0.07$ in decimal form.

STEP 4

The formula for compound interest is $A = P(1 + r)^t$ , where $A$ is the final amount, $P$ is the principal (initial amount), $r$ is the interest rate, and $t$ is the time in years.

STEP 5

Plugging in our values, we get $A = 8000(1 + 0.07)^{13}$ .

STEP 6

Calculating this, we have $A = 8000(1.07)^{13} \approx 8000 \cdot 2.409845 \approx 19278.76$ .
So, after 13 years, we'll have approximately $\$19278.76$ .

STEP 7

Now, let's tackle quarterly compounding.
Remember, quarterly means 4 times per year!

STEP 8

Our formula becomes $A = P(1 + \frac{r}{n})^{nt}$ , where $n$ is the number of times interest is compounded per year.

STEP 9

Here, $P = 8000$ , $r = 0.07$ , $n = 4$ , and $t = 13$ .
So, $A = 8000(1 + \frac{0.07}{4})^{4 \cdot 13}$ .

STEP 10

Let's break this down: $A = 8000(1 + 0.0175)^{52} = 8000(1.0175)^{52} \approx 8000 \cdot 2.483501 \approx 19868.01$ .
With quarterly compounding, we'll have approximately $\$19868.01$ .

STEP 11

Finally, let's do monthly compounding, which means $n = 12$ .

STEP 12

Our formula is $A = 8000(1 + \frac{0.07}{12})^{12 \cdot 13}$ .

STEP 13

Calculating this gives us $A = 8000(1 + \frac{0.07}{12})^{156} \approx 8000(1.005833)^{156} \approx 8000 \cdot 2.505188 \approx 20041.50$ .
So, with monthly compounding, we'll have approximately $\$20041.50$ .

STEP 14

Annually: $\$19278.76$ Quarterly: $\$19868.01$ Monthly: $\$20041.50$

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We need to figure out how much money we'll have after 13 years with different ways of compounding interest. Watch out! Don't forget to convert the percentage to a decimal!
Also, remember that compounding quarterly means 4 times a year, and monthly means 12 times a year!

STEP 2

1. Calculate annually compounded amount
2. Calculate quarterly compounded amount
3. Calculate monthly compounded amount

STEP 3

Alright, let's start with the annual compounding!
Our initial investment is $\$8000$ , and the interest rate is $7\%$ or $0.07$ in decimal form.

STEP 4

The formula for compound interest is $A = P(1 + r)^t$ , where $A$ is the final amount, $P$ is the principal (initial amount), $r$ is the interest rate, and $t$ is the time in years.

STEP 5

Plugging in our values, we get $A = 8000(1 + 0.07)^{13}$ .

STEP 6

Calculating this, we have $A = 8000(1.07)^{13} \approx 8000 \cdot 2.409845 \approx 19278.76$ .
So, after 13 years, we'll have approximately $\$19278.76$ .

STEP 7

Now, let's tackle quarterly compounding.
Remember, quarterly means 4 times per year!

STEP 8

Our formula becomes $A = P(1 + \frac{r}{n})^{nt}$ , where $n$ is the number of times interest is compounded per year.

STEP 9

Here, $P = 8000$ , $r = 0.07$ , $n = 4$ , and $t = 13$ .
So, $A = 8000(1 + \frac{0.07}{4})^{4 \cdot 13}$ .

STEP 10

Let's break this down: $A = 8000(1 + 0.0175)^{52} = 8000(1.0175)^{52} \approx 8000 \cdot 2.483501 \approx 19868.01$ .
With quarterly compounding, we'll have approximately $\$19868.01$ .

STEP 11

Finally, let's do monthly compounding, which means $n = 12$ .

STEP 12

Our formula is $A = 8000(1 + \frac{0.07}{12})^{12 \cdot 13}$ .

STEP 13

Calculating this gives us $A = 8000(1 + \frac{0.07}{12})^{156} \approx 8000(1.005833)^{156} \approx 8000 \cdot 2.505188 \approx 20041.50$ .
So, with monthly compounding, we'll have approximately $\$20041.50$ .

STEP 14

Annually: $\$19278.76$ Quarterly: $\$19868.01$ Monthly: $\$20041.50$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Calculate annually compounded amount2. Calculate quarterly compounded amount3. Calculate monthly compounded amount

STEP 3

Alright, let's start with the **annual** compounding!Our **initial investment** is $8000 \$8000 $8000, and the **interest rate** is 7% 7\% 7% or 0.07 0.07 0.07 in decimal form.

STEP 4

The formula for **compound interest** is A=P(1+r)t A = P(1 + r)^t A=P(1+r)t, where AAA is the **final amount**, PPP is the **principal** (initial amount), rrr is the **interest rate**, and ttt is the **time** in years.

STEP 5

Plugging in our values, we get A=8000(1+0.07)13 A = 8000(1 + 0.07)^{13} A=8000(1+0.07)13.

STEP 6

Calculating this, we have A=8000(1.07)13≈8000⋅2.409845≈19278.76 A = 8000(1.07)^{13} \approx 8000 \cdot 2.409845 \approx 19278.76 A=8000(1.07)13≈8000⋅2.409845≈19278.76.So, after **13 years**, we'll have approximately $19278.76\$19278.76$19278.76.

STEP 7

Now, let's tackle **quarterly** compounding.Remember, quarterly means **4 times** per year!

STEP 8

Our formula becomes A=P(1+rn)nt A = P(1 + \frac{r}{n})^{nt} A=P(1+nr​)nt, where nnn is the **number of times** interest is compounded per year.

STEP 9

Here, P=8000 P = 8000 P=8000, r=0.07 r = 0.07 r=0.07, n=4 n = 4 n=4, and t=13 t = 13 t=13.So, A=8000(1+0.074)4⋅13 A = 8000(1 + \frac{0.07}{4})^{4 \cdot 13} A=8000(1+40.07​)4⋅13.

STEP 10

STEP 11

Finally, let's do **monthly** compounding, which means n=12 n = 12 n=12.

STEP 12

Our formula is A=8000(1+0.0712)12⋅13 A = 8000(1 + \frac{0.07}{12})^{12 \cdot 13} A=8000(1+120.07​)12⋅13.

STEP 13

STEP 14

Annually: $19278.76\$19278.76$19278.76 Quarterly: $19868.01\$19868.01$19868.01 Monthly: $20041.50\$20041.50$20041.50

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate annually compounded amount
2. Calculate quarterly compounded amount
3. Calculate monthly compounded amount

Alright, let's start with the annual compounding!
Our initial investment is $\$8000$ , and the interest rate is $7\%$ or $0.07$ in decimal form.

The formula for compound interest is $A = P(1 + r)^t$ , where $A$ is the final amount, $P$ is the principal (initial amount), $r$ is the interest rate, and $t$ is the time in years.

Plugging in our values, we get $A = 8000(1 + 0.07)^{13}$ .

Calculating this, we have $A = 8000(1.07)^{13} \approx 8000 \cdot 2.409845 \approx 19278.76$ .
So, after 13 years, we'll have approximately $\$19278.76$ .

Now, let's tackle quarterly compounding.
Remember, quarterly means 4 times per year!

Our formula becomes $A = P(1 + \frac{r}{n})^{nt}$ , where $n$ is the number of times interest is compounded per year.

Here, $P = 8000$ , $r = 0.07$ , $n = 4$ , and $t = 13$ .
So, $A = 8000(1 + \frac{0.07}{4})^{4 \cdot 13}$ .

Finally, let's do monthly compounding, which means $n = 12$ .

Our formula is $A = 8000(1 + \frac{0.07}{12})^{12 \cdot 13}$ .

Annually: $\$19278.76$ Quarterly: $\$19868.01$ Monthly: $\$20041.50$