How much must be deposited at the end of each month at 6% compounded… Step by Step

Step by step

The question

Question: How much must be deposited at the end of each month at 6% compounded monthly to have $20000 after 10 years?

1 Read the numbers

FV is the money we want at the end: $20000.

We need PMT, each deposit.

r is the yearly rate: 6% is 0.06.

FV = $20,000 goal, r = 6% = 0.06

n is how many times a year: 12.

t is the time: 10 years.

n = 12 a year, t = 10 years

2 Per payment

Interest is added each month.

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

0.06 divided by 12 is 0.005.

i = rn = 0.0612 = 0.005

i = the rate each month

And N is how many deposits: 12 times 10 is 120.

N = n × t = 12 × 10 = 120

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.

20000 = PMT × 1.005¹²⁰ − 10.005

4 Work it out

With a calculator: 1.005 to the power 120 is about 1.8194.

Minus 1, divided by 0.005: about 163.879.

1.005¹²⁰ ≈ 1.8194

(1.8194 − 1) ÷ 0.005 ≈ 163.879

So 20000 is PMT times 163.879.

20000 ≈ PMT × 163.879

5 Get PMT alone

To remove times 163.879, we divide both sides by 163.879.

PMT ≈ 20000163.879

20000 divided by 163.879 is 122.0410 and a bit.

PMT ≈ 122.0410…

We round up to the next cent, so the goal is reached: 122 dollars and 5 cents.

PMT = $122.05

Careful!

Rounding down would leave the account a little short.

Careful: round UP, or the goal is not reached

6 Check

Let's check: 122 dollars and 5 cents times 163.879 is about 20001 dollars and 47 cents.

That reaches the goal.

So the answer is right!

$122.05 × 163.879 ≈ $20,001.47 ≥ $20,000

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