In this video, we'll learn two tricks for solving
compound interest problems in our heads.
Trick 1 is the Rule of 72.
A savings account pays 8% interest per year.
At the start of 2018, we deposit $1.
[♫ Mario coin sound effect. ♫]
When will this initial dollar grow to $2?
[♫ Mario coin sound effect × 2. ♫]
The Rule of 72 says that
if the interest rate is 8% per year,
the doubling time is roughly "72 divided by 8"
— or 9 years.
So, after 9 years,
our initial dollar doubles to $2.
[♫ Mario coin sound effect × 2. ♫]
After another 9 years, the $2 double to $4.
[♫ Mario coin sound effect × 4. ♫]
And after yet another 9 years, the $4 double to $8.
[♫ Mario coin sound effect × 8. ♫]
Etc.
In general, the Rule of 72 says that if a quantity
is growing at a rate of x% per time period,
the doubling time is roughly "72 divided by x" periods.
So if germs in a Petri dish are
growing at a rate of 12% per minute,
they double roughly once every "72 divided by 12",
or 6 minutes.
And if the German population is
growing at a rate of 2% per year,
it doubles roughly once every "72 divided by 2",
or 36 years.
The Rule of 72 is a rough approximation
to the actual doubling time.
If the growth rate is 8%,
it's very close to the actual doubling time.
["Very nice!"]
For lower growth rates, like 2%,
it overstates the actual doubling time.
Conversely, for higher growth rates, like 12%,
it understates the actual doubling time.
Now, why does the Rule of 72 work?
Where does the mysterious number 72 come from?
The explanation is pretty simple
but does require a bit of math.
If you're interested, see the description below.
Trick 2 is a very simple observation:
2 to the power of 10 is roughly 1,000.
That's because 2 multiplied by
itself 10 times equals 1,024.
Which is pretty close to 1,000.
That's all there is to Trick 2!
Trick 2 seems pretty useless.
But it's actually great [*slap!*] for mental arithmetic.
It tells us that growing a thousand-fold is
roughly the same as doubling 10 times.
Let's now put our two tricks to use.
"At an interest rate of 8% per year,
how long will it take for $1 to grow to $1,000?"
Trick 1 — the Rule of 72 —
says that the doubling time is roughly 9 years.
Trick 2 says that growing a thousand-fold
is roughly the same as doubling 10 times.
Altogether then, it'll take roughly 9 × 10,
or 90 years.
This rough answer is remarkably close
to the actual answer.
["Very nice!"]
Let's try another problem.
Zimbabwe is wracked by hyperinflation.
The price of an egg is initially Z$1,
but rises by 12% per day.
How long before the price hits Z$1M?
Trick 1 — the Rule of 72 —
says that the doubling time is
roughly "72 divided by 12",
or 6 days.
Now, notice that 1M is a thousand squared.
And so growing a million-fold is
the same as growing a thousand-fold TWICE.
Trick 2 says that growing a thousand-fold is
roughly the same as doubling 10 times.
So, growing a thousand-fold TWICE must be
roughly the same as doubling 20 times.
Altogether then, it'll take roughly 6 × 20,
or 120 days.
This rough answer is, again,
remarkably close to the actual answer.
["Very nice!"]
Now for the thumbnail problem.
A nasty woman borrows $1 from a loanshark,
at an interest rate of 2% per day.
Roughly how many years before her debt hits $1B?
I'll end here,
but you should be able to solve this
on your own within a minute.
Econ Cow!
Econ Cow is a new channel for educational videos,
mostly on economics.
I'm still trying to improve, so please feel free
to leave me any feedback in the comments below.
And of course, like, subscribe, and watch another video!
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