$500 grows by 4% a year. How many years until it is $800? Step by Step

Step by step

The question

Question: $500 grows by 4% a year.

How many years until it is $800?

1 Write the equation

The compound interest rule, with n = 1: once a year.

A = P(1 + rn)^nt, n = 1

We know the start, $500, and the end, $800.

t is the unknown, up in the power.

500 × 1.04^t = 800

2 Get the power alone

To remove times 500, we divide by 500.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

~~500~~ × 1.04^t~~500~~ = 800500

500 divided by 500 is 1.

So just the power is left.

800 divided by 500 is 1.6.

Now we have 1.04 to the power t = 1.6.

1.04^t = 1.6

Careful!

Each year it keeps all of it and gets 4% more.

So we multiply by 1.04, not by 0.04.

Careful: the factor is 1 + 0.04 = 1.04, not 0.04

3 Take the log of both sides

Logs bring a power down.

So we take the log of both sides.

We do it on both sides, so they stay equal.

log(1.04^t) = log 1.6

4 Bring the power down

The power rule for logs: the power comes down in front, as times.

log(aⁿ) = n × log a

The whole power, t, comes down in front.

t × log 1.04 = log 1.6

5 Get t alone

Divide both sides by log 1.04.

So t is alone.

t = log 1.6log 1.04

6 Use a calculator

With a calculator: log 1.6 is about 0.20412.

And log 1.04 is about 0.0170333.

log 1.6 ≈ 0.20412

log 1.04 ≈ 0.0170333

So t is about 11.98 years, to two decimal places.

t ≈ 11.98

7 Check

Let's check: put t = 11.98 back in.

With a calculator.

We get about $799.89.

500 × 1.04^11.98 ≈ 799.89

That is 800, up to the rounding.

It works!

≈ 800

Let's look back.

First: write the equation.

Then: get the power alone.

Then: take the log of both sides.

Then: bring the power down.

Then: get t alone.

Then: use a calculator.

Last: check.

Answer