How much must be deposited at the end of each month at 3% compounded… Step by Step

Step by step

The question

Question: How much must be deposited at the end of each month at 3% compounded monthly to have $5000 after 3 years?

1 Read the numbers

FV is the money we want at the end: $5000.

We need PMT, each deposit.

r is the yearly rate: 3% is 0.03.

FV = $5,000 goal, r = 3% = 0.03

n is how many times a year: 12.

t is the time: 3 years.

n = 12 a year, t = 3 years

2 Per payment

Interest is added each month.

The rate for one month is i: r over n.

0.03 divided by 12 is 0.0025.

i = rn = 0.0312 = 0.0025

i = the rate each month

And N is how many deposits: 12 times 3 is 36.

N = n × t = 12 × 3 = 36

3 Use the formula

The savings formula: FV = PMT times, (1 + i) to the power N, minus 1, over i.

FV = PMT × (1 + i)^N − 1i

FV = future value: the money at the end

We put the numbers in.

PMT stays a letter: we want it.

5000 = PMT × 1.0025³⁶ − 10.0025

4 Work it out

With a calculator: 1.0025 to the power 36 is about 1.09405.

Minus 1, divided by 0.0025: about 37.6206.

1.0025³⁶ ≈ 1.09405

(1.09405 − 1) ÷ 0.0025 ≈ 37.6206

So 5000 is PMT times 37.6206.

5000 ≈ PMT × 37.6206

5 Get PMT alone

To remove times 37.6206, we divide both sides by 37.6206.

PMT ≈ 500037.6206

5000 divided by 37.6206 is 132.9060 and a bit.

PMT ≈ 132.9060…

We round up to the next cent, so the goal is reached: 132 dollars and 91 cents.

PMT = $132.91

Careful!

Rounding down would leave the account a little short.

Careful: round UP, or the goal is not reached

6 Check

Let's check: 132 dollars and 91 cents times 37.6206 is about 5000 dollars and 15 cents.

That reaches the goal.

So the answer is right!

$132.91 × 37.6206 ≈ $5,000.15 ≥ $5,000

Let's look back.

First: read the numbers.

Then: per payment.

Then: use the formula.

Then: work it out.

Then: get PMT alone.

Last: check.

Answer