Rates

Rates compare quantities with different units and make speed, price, density, productivity, and proportional change measurable.

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Equivalent ratios scale both quantities by the same factor. A ten-by-ten grid, a double number line, and paired arrows show the same proportional relationship in percent, fraction, and rate form.

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

\frac{180\text{ km}}{3\text{ h}}=60\text{ km/h}.

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

\frac{\$20}{8\text{ notebooks}}=\$2.50/\text{notebook}.

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

A dependable method

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

Solving proportional rate problems

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

$$ y=rx. $$

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.

Converting units

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

72\frac{\text{km}}{\text{h}}\times \frac{1000\text{ m}}{1\text{ km}}\times \frac{1\text{ h}}{3600\text{ s}} =20\frac{\text{m}}{\text{s}}.

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

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.