Compound Interest | Math Coach Amy

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Key Ideas Easy Medium Hard

SL 1.4

$FV = PV \times \left(1 + \dfrac{r}{100k}\right)^{kn}$

What to Remember

Compound interest

Each compounding period multiplies the balance by the same factor. The balances form a geometric sequence.

The formula booklet gives

$FV = PV \times \left(1 + \dfrac{r}{100k}\right)^{kn}$

FV is the future value, PV is the present value, n is the number of years, k is the number of times interest compounds per year, and r% is the annual interest rate.

TI-84 Plus CE

Press APPS, then 1: Finance, then 1: TVM Solver.

N is the total number of compounding periods, not always the number of years.

I% is the annual rate. Type 4.8 for 4.8%. Do not divide by 12 yourself.

PV is the starting amount. Make it negative with the (−) key. The subtraction key does not make a negative here.

PMT is 0 when there is no regular payment.

FV is the value you want. Move the cursor to FV and press ALPHA ENTER. ALPHA ENTER is SOLVE, printed above ENTER.

P/Y and C/Y are the number of times per year. They stay equal. Type P/Y and press ENTER, and C/Y changes to match.

Leave PMT: END selected.

If the calculator says ERR: DOMAIN, PV is not negative.

Leave the mode in FLOAT. Round money to the nearest cent. If the screen already shows two decimals, the mode is FIX 2 and that display is the rounded cent value.

Worked example

$4,000 is invested at 4.8% annual interest, compounded monthly, for 15 months.

$k = 12$

$n = \dfrac{15}{12}$

$kn = 15$

$FV = 4000 \times \left(1 + \dfrac{4.8}{100 \times 12}\right)^{15}$

On the solver, N is 15 and I% stays 4.8. P/Y and C/Y are 12.

N = 15
I% = 4.8
PV = −4000
PMT = 0
P/Y = 12
C/Y = 12
PMT: END

TI-84 Plus CE TVM Solver. N=15, I%=4.8, PV=-4000, PMT=0, FV=4246.83789014 with a solve mark, P/Y=12, C/Y=12, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 4246.83789014$

To the nearest cent, the balance is $4,246.84.

Easy

Annual Compounding

Interest is compounded once a year. Find the balance.

Example

$2,500 is invested at 4% annual interest, compounded annually. Find the balance after 6 years.

Enter the amount invested as a negative PV.

Interest is compounded once a year for 6 years, so N = 6.

N = 6
I% = 4
PV = −2500
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=6, I%=4, PV=-2500, PMT=0, FV=3163.29754624 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 3163.29754624$

To the nearest cent, the balance is $3,163.30.

Answer: $3,163.30

$1,800 is invested at 5% annual interest, compounded annually. Find the balance after 4 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded once a year for 4 years, so N = 4.

N = 4
I% = 5
PV = −1800
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=4, I%=5, PV=-1800, PMT=0, FV=2187.91125 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 2187.91125$

To the nearest cent, the balance is $2,187.91.

Answer: $2,187.91

$3,200 is invested at 3% annual interest, compounded annually. Find the balance after 8 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded once a year for 8 years, so N = 8.

N = 8
I% = 3
PV = −3200
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=8, I%=3, PV=-3200, PMT=0, FV=4053.66426044 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 4053.66426044$

To the nearest cent, the balance is $4,053.66.

Answer: $4,053.66

$750 is invested at 6% annual interest, compounded annually. Find the balance after 5 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded once a year for 5 years, so N = 5.

N = 5
I% = 6
PV = −750
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=5, I%=6, PV=-750, PMT=0, FV=1003.6691832 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 1003.6691832$

To the nearest cent, the balance is $1,003.67.

Answer: $1,003.67

$4,600 is invested at 2% annual interest, compounded annually. Find the balance after 10 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded once a year for 10 years, so N = 10.

N = 10
I% = 2
PV = −4600
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=10, I%=2, PV=-4600, PMT=0, FV=5607.37433198 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 5607.37433198$

To the nearest cent, the balance is $5,607.37.

Answer: $5,607.37

$910 is invested at 7% annual interest, compounded annually. Find the balance after 3 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded once a year for 3 years, so N = 3.

N = 3
I% = 7
PV = −910
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=3, I%=7, PV=-910, PMT=0, FV=1114.78913 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 1114.78913$

To the nearest cent, the balance is $1,114.79.

Answer: $1,114.79

Medium

Monthly and Quarterly

Interest compounds more than once a year. N is the total number of periods. I% stays the annual rate.

Example

$1,500 is invested at 6% annual interest, compounded quarterly. Find the balance after 5 years.

Enter the amount invested as a negative PV.

Interest is compounded quarterly for 5 years, so N = 4 × 5 = 20. I% stays the annual rate.

N = 20
I% = 6
PV = −1500
PMT = 0
P/Y = 4
C/Y = 4
PMT: END

TI-84 Plus CE TVM Solver. N=20, I%=6, PV=-1500, PMT=0, FV=2020.28250983 with a solve mark, P/Y=4, C/Y=4, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 2020.28250983$

To the nearest cent, the balance is $2,020.28.

Answer: $2,020.28

$2,400 is invested at 3.6% annual interest, compounded monthly. Find the balance after 2 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded monthly for 2 years, so N = 12 × 2 = 24. I% stays the annual rate.

N = 24
I% = 3.6
PV = −2400
PMT = 0
P/Y = 12
C/Y = 12
PMT: END

TI-84 Plus CE TVM Solver. N=24, I%=3.6, PV=-2400, PMT=0, FV=2578.89484592 with a solve mark, P/Y=12, C/Y=12, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 2578.89484592$

To the nearest cent, the balance is $2,578.89.

Answer: $2,578.89

$800 is invested at 5% annual interest, compounded quarterly. Find the balance after 3 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded quarterly for 3 years, so N = 4 × 3 = 12. I% stays the annual rate.

N = 12
I% = 5
PV = −800
PMT = 0
P/Y = 4
C/Y = 4
PMT: END

TI-84 Plus CE TVM Solver. N=12, I%=5, PV=-800, PMT=0, FV=928.60361418 with a solve mark, P/Y=4, C/Y=4, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 928.60361418$

To the nearest cent, the balance is $928.60.

Answer: $928.60

$2,750 is invested at 3.6% annual interest, compounded monthly. Find the balance after 20 months.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded monthly for 20 months, so N = 20. I% stays the annual rate.

N = 20
I% = 3.6
PV = −2750
PMT = 0
P/Y = 12
C/Y = 12
PMT: END

TI-84 Plus CE TVM Solver. N=20, I%=3.6, PV=-2750, PMT=0, FV=2919.78823466 with a solve mark, P/Y=12, C/Y=12, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 2919.78823466$

To the nearest cent, the balance is $2,919.79.

Answer: $2,919.79

$650 is invested at 8% annual interest, compounded twice a year. Find the balance after 4 years.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded twice a year for 4 years, so N = 2 × 4 = 8. I% stays the annual rate.

N = 8
I% = 8
PV = −650
PMT = 0
P/Y = 2
C/Y = 2
PMT: END

TI-84 Plus CE TVM Solver. N=8, I%=8, PV=-650, PMT=0, FV=889.56988276 with a solve mark, P/Y=2, C/Y=2, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 889.56988276$

To the nearest cent, the balance is $889.57.

Answer: $889.57

$12,000 is invested at 2.4% annual interest, compounded monthly. Find the balance after 30 months.

Show Answer

Enter the amount invested as a negative PV.

Interest is compounded monthly for 30 months, so N = 30. I% stays the annual rate.

N = 30
I% = 2.4
PV = −12000
PMT = 0
P/Y = 12
C/Y = 12
PMT: END

TI-84 Plus CE TVM Solver. N=30, I%=2.4, PV=-12000, PMT=0, FV=12741.27507694 with a solve mark, P/Y=12, C/Y=12, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 12741.27507694$

To the nearest cent, the balance is $12,741.28.

Answer: $12,741.28

Hard

Loans With No Payments

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Example

A loan of $7,000 charges 6% annual interest, compounded annually. No payments are made for 5 years. How much must be paid at the end?

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Interest is compounded once a year for 5 years, so N = 5.

N = 5
I% = 6
PV = −7000
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=5, I%=6, PV=-7000, PMT=0, FV=9367.5790432 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 9367.5790432$

To the nearest cent, the amount to pay is $9,367.58.

Answer: $9,367.58

A loan of $4,500 charges 4% annual interest, compounded annually. No payments are made for 6 years. How much must be paid at the end?

Show Answer

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Interest is compounded once a year for 6 years, so N = 6.

N = 6
I% = 4
PV = −4500
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=6, I%=4, PV=-4500, PMT=0, FV=5693.93558323 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 5693.93558323$

To the nearest cent, the amount to pay is $5,693.94.

Answer: $5,693.94

A loan of $12,500 charges 3.5% annual interest, compounded annually. No payments are made for 8 years. How much must be paid at the end?

Show Answer

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Interest is compounded once a year for 8 years, so N = 8.

N = 8
I% = 3.5
PV = −12500
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=8, I%=3.5, PV=-12500, PMT=0, FV=16460.11296204 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 16460.11296204$

To the nearest cent, the amount to pay is $16,460.11.

Answer: $16,460.11

A loan of $8,200 charges 7% annual interest, compounded monthly. No payments are made for 2 years. How much must be paid at the end?

Show Answer

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Interest is compounded monthly for 2 years, so N = 12 × 2 = 24. I% stays the annual rate.

N = 24
I% = 7
PV = −8200
PMT = 0
P/Y = 12
C/Y = 12
PMT: END

TI-84 Plus CE TVM Solver. N=24, I%=7, PV=-8200, PMT=0, FV=9428.40934352 with a solve mark, P/Y=12, C/Y=12, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 9428.40934352$

To the nearest cent, the amount to pay is $9,428.41.

Answer: $9,428.41

A loan of $3,000 charges 5.5% annual interest, compounded quarterly. No payments are made for 3 years. How much must be paid at the end?

Show Answer

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Interest is compounded quarterly for 3 years, so N = 4 × 3 = 12. I% stays the annual rate.

N = 12
I% = 5.5
PV = −3000
PMT = 0
P/Y = 4
C/Y = 4
PMT: END

TI-84 Plus CE TVM Solver. N=12, I%=5.5, PV=-3000, PMT=0, FV=3534.20438469 with a solve mark, P/Y=4, C/Y=4, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 3534.20438469$

To the nearest cent, the amount to pay is $3,534.20.

Answer: $3,534.20

A loan of $9,600 charges 2.8% annual interest, compounded annually. No payments are made for 10 years. How much must be paid at the end?

Show Answer

Enter the amount borrowed as a negative PV. The positive FV is the amount to pay back.

Interest is compounded once a year for 10 years, so N = 10.

N = 10
I% = 2.8
PV = −9600
PMT = 0
P/Y = 1
C/Y = 1
PMT: END

TI-84 Plus CE TVM Solver. N=10, I%=2.8, PV=-9600, PMT=0, FV=12653.25847302 with a solve mark, P/Y=1, C/Y=1, END selected.

Move the cursor to FV. Press ALPHA ENTER.

The calculator shows

$FV = 12653.25847302$

To the nearest cent, the amount to pay is $12,653.26.

Answer: $12,653.26

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