EQN-02-03

Unit Rates, Speed & Best Value

Reduce any rate to "per one" and you can compare prices, speeds, and productivity instantly.

~5 min read•Difficulty 2/5

A rate compares two different units (km per hour, price per kg). A unit rate has 1 in the denominator. Divide to get it.

240 km in 4 hours \to 240 \div 4 = 60 km per hour

speed = distance / time distance = speed \times time time = distance / speed

Keep units consistent. If speed is km/h, time must be in hours (30 minutes $= 0.5 h) .$

Average speed is total distance $\div$ total time — not the average of the two speeds, unless the times are equal.

120 km at 60 km / h (2 h) then 120 km at 40 km / h (3 h) :

average = 240/5 = 48 km / h (not 50)

To compare deals, reduce each to price per single unit; the smallest unit price wins.

Worked examples

Example 1 (difficulty 2/5)

Which is the better value: 6 pens for 9, or 10 pens for 14?

◎ Predict the answer first:

Example 2 (difficulty 1/5)

A train covers 150 km in 2.5 hours. What is its average speed?

◎ Predict the answer first:

Reduce everything to "per one" before comparing — rates become directly comparable.
Convert minutes to hours $(\div 60)$ before using km/h, or units will not match.
Average speed $=$ total distance $\div$ total time, never the mean of the speeds.

Averaging two speeds directly to get average speed

Why: It seems symmetric.

Fix: Use total distance over total time; equal speeds average only when the times are equal.

Mixing minutes and hours in a speed calculation

Why: The numbers are used as given.

Fix: Make the time unit match the speed unit first.