Key takeaway: A rate compares two amounts that have different units, such as 45 gallons in 6 minutes, and is usually read with the word per, as in gallons per minute.
Grade 66.RP.A.2Ratios, Rates and Percents
Definition of Rate
A rate sets two amounts side by side when their units differ. A garden hose that puts 45 gallons into a tub in 6 minutes gives the pair 45 gallons for 6 minutes. Speed, hourly pay and cost per item all work this way.
The word per means “for each”. Since the units differ, the comparison carries both of them: miles per hour, dollars per pound, words per minute. When the second amount is exactly 1, you have a unit rate, which is the easiest form to compare.
Formula and notation
Write it as a fraction with units, with the word per, or with a slash:
\[\text{rate} = \frac{45\text{ gallons}}{6\text{ minutes}}\]
Dividing gives 7.5 gallons per minute, also written \(7.5\text{ gal/min}\). The unit that follows per always goes in the denominator.
Key facts
- Two different units separate a rate from a ratio whose terms share one unit.
- Dividing the top amount by the bottom amount gives the quantity for exactly one unit.
- When the pace stays steady, a graph of the two amounts is a straight line through the origin.
- Speed is distance per unit of time; multiplying speed by time gives distance.
Rate example
A garden hose fills a 45-gallon tub in 6 minutes. Find how many gallons flow each minute, then the amount of water after 20 minutes at that pace.
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Write the comparison: \(\frac{45\text{ gallons}}{6\text{ minutes}}\).
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Divide: \(45 : 6 = 7.5\) gallons per minute.
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Multiply by the time: \(7.5 \times 20 = 150\) gallons.
The hose delivers 7.5 gallons per minute, so 150 gallons flow in 20 minutes.
Common mistake
Watch out
Maya bikes 12 miles in 45 minutes and writes her rate as \(12 : 45 \approx 0.27\) miles per hour.
Why it is wrong
The 45 is in minutes, so the quotient is in miles per minute, not per hour. Convert first: 45 minutes is \(0.75\) hour, and \(12 : 0.75 = 16\) miles per hour. Check that the unit in the denominator matches the unit named after per.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Ratios, Rates and Percents