Here’s a collection of time-saving math shortcuts, great for back-of-the-envelope estimates.

## Time and Distance

**60 mph = 1 mile per minute**

- Going 60 mph and the exit is in 10 miles? That’s 10 minutes.
- Been driving a half hour? That’s about 30 miles at highway speeds.

**Feet Per Second = MPH * 1.5**

**MPH = Feet Per Second * 2/3** (derivation)

- 60 mph is about 90 feet per second (88 exactly), so just multiply by 1.5. Or, just add half to itself (60 + 30 = 90).
- Going 100 mph? That’s 150 fps.
- Going 10 fps? That’s about 7 mph (10 * 2/3 is 6.666). Or, just take away 1/3 (10 – 3 = 7).

**speed of light = 1 foot per nanosecond** (derivation)

- The US is about 3000 miles long. There’s about 5000 feet/mile, so that’s about 3000 × 5000 or 15 million feet. 15 million feet takes 15 million nanoseconds, or 15/1000, or 15 milliseconds. That’s the minimum time for a signal to go across the country.
- Inside a microchip, if you have a clock cycle every nanosecond (1 GHz), your signal can only travel 1 foot at most (or less, depending on the material). Even light takes 30ns to cross a 30 foot room.

**1 year = 250 work days = 2000 work hours** (derivation)

- Project takes 1000 man hours? That’s 6 months for 1 person.
- Daily commute of 1/2 hour? That’s .5 * 250 = 125 hours in the car each year.

## Money and Finance

**$1/hour = $2000/year** (derivation)

- Earn $25/hour? That’s about 50k/year.
- Make 200k/year? That’s about $100/hour. This assumes a 40-hour work week.

**$20/week = $1000/year** (derivation)

- Spend $20/week at Starbucks? That’s a cool grand a year.

**Rule of 72: Years To Double = 72/Interest Rate** (derivation)

- Have an investment growing at 10% interest? It will double in 7.2 years.
- Want your investment to double in 5 years? You need 72/5 or about 15% interest.
- Growing at 2% a week? You’ll double in 72/2 or 36 weeks. You can use this rule for any duration of time, not just years.
- Inflation at 4%? It will halve your money in 72/4 or 18 years.

## Mental Arithmetic

**Numbers**

**10,000 = hundred hundred**

**million = thousand thousand**

**billion = thousand million**

**trillion = million million**

- 1% of 10k is 100. The Dow is roughly 10k (it’s about 12k now). So if the dow drops 100, it’s about a 1% loss.
- What’s 5k x 50k? That’s 250 * thousand * thousand or 250 million.

**Visualizing numbers (read more)**

- 12 days = 1 million seconds
- 1 year = 31 million seconds (about pi * 10 million)
- 30 years = 1 billion seconds
30,000 years = 1 trillion seconds

One “part per million” means an accuracy of 1 second every 12 days. One “part per trillion” means an accuracy of 1 second every 30,000 years.

**Powers of 2**

**2^6 = 64 (the sixes match: six and sixty-four)**

**2^10 ~ thousand** (1 kb)

**2^20 ~ million** (1 mb)

**2^30 ~ billion** (1 gb)

- Sure, 2 to the tenth = 1024, but 1000 is good enough for government work. (Read on about KB vs KiB).
- Have 32-bit color? That’s 2 + 30 bits = 2^2 * 2^30 = 2^2 billion = 4 billion (4gb exactly).
- Have a 16-bit number? That’s 6 + 10 bits, or 2^6 thousand, or 64 thousand (64 kb).

**a% of b = b% of a**

- It’s not immediately clear, but it’s true: a% of b = .01 * a * b, which is the same as b% of a (.01 * b * a).
- What’s 16% of 25? The same as 25% of 16: 4
- What’s 43% of 200? Same as 200% of 43: 86.

## Leave a Reply

123 Comments on "Mental Math Shortcuts"

nice blog…

[…] Mental Math Shortcuts | BetterExplained Heres a collection of time-saving math shortcuts, great for back-of-the-envelope estimates. (tags: math) […]

Two words: Metric System.

I like the metric system as much as the next guy, but 1 foot per nanosecond just works out so nicely :)

Speed of light in metric: 3 decimeters per nanosecond (source). For those who don’t know, a decimeter is 1/10 of a meter; thus, another way of saying it would be 0.3 meters per nanosecond.

[…] Mental Math Shortcuts […]

[…] Let’s say you want to add the numbers from 1 to 1000: suppose you get 1 additional visitor to your site each day – how many total visitors will you have after 1000 days? Since thousand squared = 1 million, we get […]

[…] A single byte has 8 bits, or 2^8 (256) possible values. 4 bytes gives us 2^32 bits, or roughly 4 billion values. So, we could store the number 4 billion in only 4 bytes. […]

whew! nice math shortcuts!

[…] The Rule of 72 is a mental math shortcut to estimate the time needed to double your money. We’re going to derive it (yay!) and even better, we’re going to understand it intuitively. […]

This line is not correct:

10,000 = hundred hundred

how to multiply in shortcuts

Hi Steven, feel free try it out: 100 * 100 = 10,000 (hundred hundred is similar to “two hundred (200)”, “fifteen hundred (1500)”, “forty-seven hundred (4700)” or “hundred hundred (10,000)”. We don’t often say “hundred hundred” though :).

Gladys/Jean, I may do a follow up with mental multiplication tricks.

This line is not correct:10,000 = hundred hundred

yes it is

Electrical signals in semiconductors do not travel at the speed of light. Your 1GHz clock distance

calculation is wrong.

Hi John, thanks for the info. Yes, it seems that the speed of electricity varies depending on the conductor.

http://howthingswork.virginia.edu/page1.php?QNum=1267

If there’s a factor of .66c, then it would be 8 inches per nanosecond (rather than 1 foot). I’ll update the article.

two more words: METRIC SYSTEM

I love the metric system too, but 1 foot per nanosecond just works out well, don’t you think?

30 centimeters per second doesn’t have quite the same ring to it :).

It’s misleading to say that a 16 bit number somehow equates to 64kb. A 16 bit number can REPRESENT any of 64 thousand different integers. But it is only made up of TWO bytes (8 bit bytes).

Good point. a 16-bit number can address up to 64Kb of memory, but doesn’t take 64kb of room. I’ll clarify.

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Kalid, great site! In your derivation for work hours, the third line should read:

days per year = weeks * days

Cheers!

PS Perhaps, in a future article you might be inclined to explain UNIX load averages (and different kinds of averages in general, like here)?

Hi Marc, thanks for comment & catch — should be fixed now! That’s a good topic suggestion, I didn’t realize there could be so many intricacies in a “simple” performance metric :)

The last paragraphs discussing 2^10, 2^20, etc, may be a good place to introduce the concept of KiB, MiB, GiB nomenclature, (http://en.wikipedia.org/wiki/Kibibyte)

Hi, thanks for the suggestion. I think I’ll put in a link for the “new” notation.