9.4Rates and Unit Rates

A rate compares two quantities of different kinds. A unit rate gives the amount per single unit of the second quantity, which makes comparing different rates fair and straightforward.

Common rates: speed (km/h), price (dollars per kg), water flow (litres/min), wages (dollars per hour)
To compare rates fairly, convert both to unit rates first
The better buy has the lower unit price; the faster worker has the higher unit output

Finding a unit rate to compare deals

A hawker stall sells chicken rice. Small pack: 3 pieces of chicken for $\displaystyle \$5.40$ . Large pack: 5 pieces for $\displaystyle \$8.50$ . Which pack gives better value?

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\displaystyle \text{Small pack unit price}

\displaystyle = \dfrac{\$5.40}{3}

\displaystyle = \$1.80 \text{ per piece}

\displaystyle \text{Large pack unit price}

\displaystyle = \dfrac{\$8.50}{5}

\displaystyle = \$1.70 \text{ per piece}

The large pack gives better value (lower unit price)

Rate involving water flow

A garden sprinkler delivers 36 litres of water in 9 minutes. A second sprinkler delivers 55 litres in 11 minutes. Which sprinkler has the higher flow rate?

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\displaystyle \text{Sprinkler 1 flow rate}

\displaystyle = \dfrac{36}{9}

\displaystyle = 4 \text{ litres/min}

\displaystyle \text{Sprinkler 2 flow rate}

\displaystyle = \dfrac{55}{11}

\displaystyle = 5 \text{ litres/min}

Sprinkler 2 has the higher flow rate.

Now you try

A worker earns $\$720$ in $6$ days. Find the daily rate of pay.

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\displaystyle \text{Daily rate}

\displaystyle = \frac{\$720}{6 \text{ days}}

\displaystyle = \$120 \text{ per day}

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