Rates and Speed

Rates #

How much of one thing per one of another.

A rate tells you how much of one thing goes with one of another

Every hour the tap adds the same 12 liters. The mark above 1 is the rate.

3 hours is 3 of those steps: 3 × 12 = 36 liters, read off underneath.

Rate, Total, Units #

Any two of the three find the third.

Knowing any two of rate, total and number of units gives you the third

5 each time, 3 times over: 3 × 5 = 15 altogether.

The same 15 as 3 equal parts of 5. Rate × units = total: 5 × 3 = 15.

Cover the rate and divide it back out: 15 ÷ 3 = 5 in each part.

Cover the count instead: 15 ÷ 5 = 3 parts. Any two of them find the missing one.

Comparing Unit Rates #

Cost per item shows the better buy.

Cost per item is what makes two different-sized packs comparable

12 buys 4 here and 6 there. You cannot compare them yet.

12 ÷ 4 = 3 each, against 12 ÷ 6 = 2 each. The cheaper buy is now clear.

Exchange Rates #

One money priced in another.

An exchange rate prices one unit of a currency, and converts like any other rate

Every dollar buys the same 1.5 euros — a rate, like liters per hour.

Toward euros, multiply: $4 is 4 × 1.5 = 6 euros.

Coming back divides by the same 1.5 — the trap is multiplying both ways.

Speed #

Distance shared out over time.

Speed is distance shared out over time

120 km traveled, and the trip took 4 hours.

Share the 120 km between the 4 hours: 120 ÷ 4 = 30 km every hour.

Average Speed #

Whole distance over whole time.

Average speed is the whole distance over the whole time, never the mean of the speeds

Two equal legs of 60 km: the fast one takes 1 h, the slow one takes 2 h.

Average speed reads the whole trip at once: 120 km over 3 h is 40 km/h.

The mean of the speeds says 45 — wrong, because the slow leg lasts twice as long.

Converting Compound Units #

Kilometers per hour into meters per second.

A compound unit converts both of its parts, the one on top and the one underneath

72 km/h is 72 km over 1 hour — numerator and denominator, each with a unit.

Change the numerator first: 72 km is 72000 m, so 72000 m in one hour.

Now the denominator: one hour is 3600 s, and $72000/3600=20\frac{m}{s}$72000 / 3600 = 20 m/s.

$20\frac{m}{s}$20 m/s is 20 m every second — the same motion, measured a different way.

Every km/h takes those same two steps, so km/h to $\frac{m}{s}$m/s is a divide by 3.6.

Density #

How much is packed into each cubic centimeter.

Density is how much mass is packed into each unit of space

A 4 cm³ block weighing 12 g packs 3 g into each cubic centimeter.

Mass ÷ volume, the same shape as speed: an amount spread over each unit.

The rate recovers the rest: 40 cm³ of the same material weighs 3 × 40 = 120 g.

Pressure #

Force spread over the area it presses on.

Pressure is the force spread over each unit of the area it presses on

A 12 N push spread over 4 cm² presses 3 N onto each square centimeter.

Squeeze the same 12 N onto 2 cm² and each square centimeter takes 6 N.

Force ÷ area, the same shape as density: an amount spread over each unit.

It runs back too: 3 N on each of 4 cm² comes to 3 × 4 = 12 N in all.