Rates

A rate is a ratio that compares quantities measured in different units, such as kilometres per hour or dollars per kilogram.

What “per” means

The word “per” signals division. A vehicle travelling $180$ km in $3$ h has an average rate

Units are part of the mathematics. The number $60$ alone does not reveal whether it describes speed, price, flow, or something else.

Unit rates

A unit rate has a denominator quantity of $1$. It makes comparisons fair when original quantities differ. If $8$ notebooks cost $20$ dollars, the unit price is

The reciprocal rate, $0.4$ notebooks per dollar, is also valid but answers a different question.

A dependable method

1. Name the output and input quantities.
2. Write the requested unit order.
3. Divide matching quantities in that order.
4. Simplify to a denominator of $1$ when a unit rate is useful.
5. Keep units visible and check whether the size is reasonable.

Solving proportional rate problems

If a rate remains constant, output equals rate times input:

At $60$ km/h, distance after $2.5$ h is $60(2.5)=150$ km. Rearranging the same relationship gives $r=y/x$ and $x=y/r$.

Constant-rate assumptions matter. A real trip may include traffic and stops, so $60$ km/h could be an average rather than the speed at every moment.

Comparing rates

Convert rates to the same units before deciding which is greater. A price of $6.60$ per $2$ kg is $3.30$/kg. A price of $5.20$ per $1.5$ kg is about $3.47$/kg, so the first option has the lower unit price.

“Best” can depend on more than price—quality, waste, and the amount actually needed may matter—but the unit rate gives a clean numerical comparison.

Converting units

Use conversion factors equal to $1$ so unwanted units cancel. To convert $72$ km/h to m/s:

Write units as carefully as numerical factors; they show whether the conversion is oriented correctly.

Rates on graphs

For a proportional relationship, the graph is a line through the origin and the unit rate is its slope. If cost $C=2.5n$, the slope $2.5$ means $2.50$ dollars per item. A line that does not pass through the origin may include a fixed starting amount, such as a delivery fee.

Common mistakes

Dividing in the wrong order. Decide whether the question asks kilometres per hour or hours per kilometre.

Comparing unlike units. Dollars per package cannot be compared directly when package sizes differ.

Dropping the units. Units are the meaning of a rate and help expose reversed calculations.

Assuming every relationship is proportional. A fixed fee, changing speed, or bulk discount can make the rate vary.

Quick self-check

• What does one unit of the denominator quantity correspond to?
• Are both rates expressed with matching units?
• Does the context support a constant-rate model?
• Can I multiply my unit rate back to recover the original quantity?

Related topics

• Ratios
• Proportions
• Percent
• Slope