How much must be deposited at the end of each month at 2.4%… Step by Step

Step by step

The question

Question: How much must be deposited at the end of each month at 2.4% compounded monthly to have $12000 after 2 years?

1 Read the numbers

FV is the money we want at the end: $12000.

We need PMT, each deposit.

r is the yearly rate: 2.4% is 0.024.

FV = $12,000 goal, r = 2.4% = 0.024

n is how many times a year: 12.

t is the time: 2 years.

n = 12 a year, t = 2 years

2 Per payment

Interest is added each month.

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

0.024 divided by 12 is 0.002.

i = rn = 0.02412 = 0.002

i = the rate each month

And N is how many deposits: 12 times 2 is 24.

N = n × t = 12 × 2 = 24

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.

12000 = PMT × 1.002²⁴ − 10.002

4 Work it out

With a calculator: 1.002 to the power 24 is about 1.04912.

Minus 1, divided by 0.002: about 24.5602.

1.002²⁴ ≈ 1.04912

(1.04912 − 1) ÷ 0.002 ≈ 24.5602

So 12000 is PMT times 24.5602.

12000 ≈ PMT × 24.5602

5 Get PMT alone

To remove times 24.5602, we divide both sides by 24.5602.

PMT ≈ 1200024.5602

12000 divided by 24.5602 is 488.5957 and a bit.

PMT ≈ 488.5957…

We round up to the next cent, so the goal is reached: 488 dollars and 60 cents.

PMT = $488.60

Careful!

Rounding down would leave the account a little short.

Careful: round UP, or the goal is not reached

6 Check

Let's check: 488 dollars and 60 cents times 24.5602 is about 12000 dollars and 10 cents.

That reaches the goal.

So the answer is right!

$488.60 × 24.5602 ≈ $12,000.10 ≥ $12,000

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