$800 is invested at 4% compounded quarterly. How long until it… Step by Step

Step by step

The question

Question: $800 is invested at 4% compounded quarterly.

How long until it doubles?

1 Write the equation

The compound interest rule, with r/n = 0.04 divided by 4: 0.01.

A = P(1 + rn)^nt, 0.04 ÷ 4 = 0.01

We know the start, $800, and the end, $1600.

t is the unknown, up in the power.

800 × 1.01^4t = 1600

2 Get the power alone

To remove times 800, we divide by 800.

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.

~~800~~ × 1.01^4t~~800~~ = 1600800

800 divided by 800 is 1.

So just the power is left.

1600 divided by 800 is 2.

Now we have 1.01 to the power 4t = 2.

1.01^4t = 2

Careful!

When the power comes down, all of it comes down: 4t, not just t.

Careful: the whole power 4t comes down, not only t

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(1.01^4t) = log 2

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, 4t, comes down in front.

4t × log 1.01 = log 2

5 Get t alone

Divide both sides by 4 times log 1.01.

So t is alone.

t = log 24 × log 1.01

6 Use a calculator

With a calculator: log 2 is about 0.30103.

And log 1.01 is about 0.00432137.

log 2 ≈ 0.30103

log 1.01 ≈ 0.00432137

So t is about 17.42 years, to two decimal places.

t ≈ 17.42

7 Check

Let's check: put t = 17.42 back in.

With a calculator.

We get about $1600.31.

800 × 1.01^{4 × 17.42} ≈ 1600.31

That is 1600, up to the rounding.

It works!

≈ 1600

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