The question
Question: A substance has a half-life of 5 years.
How much of 80 g is left after 20 years?
1 Count the half-lives
Every 5 years, half is left.
Half means times 1/2.
every 5 years: × 12
How many half-lives in 20 years?
20 divided by 5: 4.
half-lives = 205 = 4
Careful!
4 half-lives is not divided by 4.
It is halved 4 times, one after another.
Careful: not 80 ÷ 4 = 20 g ✗
2 Halve it that many times
A = the start times (1/2) to the power 4.
A = 80 × (12)4
(1/2) to the power 4 is 1/16.
(12)4 = 116
80 times 1/16 is 80/16.
A = 8016
80 divided by 16 is 5 g.
A = 5 g
3 Check
Let's check, half-life by half-life.
Every 5 years, we halve it.
After 20 years we get 5 g again.
It works!
after 20 years: 5 g
Let's look back.
First: count the half-lives.
Then: halve it that many times.
Last: check.
▶ Watch it solved
The question
Question: A substance has a half-life of 6 years.
How much of 120 g is left after 24 years?
1 Count the half-lives
Every 6 years, half is left.
Half means times 1/2.
every 6 years: × 12
How many half-lives in 24 years?
24 divided by 6: 4.
half-lives = 246 = 4
Careful!
4 half-lives is not divided by 4.
It is halved 4 times, one after another.
Careful: not 120 ÷ 4 = 30 g ✗
2 Halve it that many times
A = the start times (1/2) to the power 4.
A = 120 × (12)4
(1/2) to the power 4 is 1/16.
(12)4 = 116
120 times 1/16 is 120/16.
A = 12016
120 divided by 16 is 7.5 g.
A = 7.5 g
3 Check
Let's check, half-life by half-life.
Every 6 years, we halve it.
After 24 years we get 7.5 g again.
It works!
after 24 years: 7.5 g
Let's look back.
First: count the half-lives.
Then: halve it that many times.
Last: check.
▶ Watch it solved
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 × 1.014t = 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 × 1.014t800 = 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.
1.014t = 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(1.014t) = 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 × 1.014 × 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.
▶ Watch it solved
