Mental Math Shortcuts

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.

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