Sizu Riatto
Calculus · Chapter 2

Rate Flipper

A rate is one change divided by another, and its units say so. Every rate can be flipped, and sometimes the flipped one tells the truth more plainly.

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Opens on Chapter 2’s example: a 10-minute mile and a 5K; 25 against 40 mpg over 10,000 miles; a 1.5 kW heater for 6 hours at 16¢; a 4.5 GB download on 100 Mbps; 120 cups in 3 hours.

1. Pace and speed
2. Miles per gallon, flipped
3. Power and energy
4. A download
5. How long a supply lasts
Enter your numbers, then press Work it out.

How it works

A rate's units are a division: miles per hour is miles ÷ hours. Flip it and you get hours per mile, the pace. Miles per gallon flipped is gallons per mile, and gallons per 100 miles shows savings honestly: going from 10 to 20 mpg saves more fuel than going from 25 to 40. Kilowatts are a rate of using energy; kilowatt-hours are the amount (kW × hours), and the bill charges for the amount. Download speeds are in megabits per second while files are in bytes: there are 8 bits in a byte (here 1 GB = 1,000 MB). A supply lasts amount ÷ rate.

From the book, Chapter 2. A 10-minute mile is 6 mph, and a 5K (3.11 miles) takes about 31 minutes. Over 10,000 miles, 25 mpg burns 400 gallons and 40 mpg 250, saving 150; 10 to 20 mpg saves 500. The heater uses 9 kWh, $1.44. The download is 36,000 megabits, 360 seconds = 6 minutes. 120 cups in 3 hours is 40 an hour, so a 200-cup urn lasts 5 hours. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.