Rates
Lesson · Beginner
Arithmetic
A rate compares how much of something is completed with the amount of time required.
If units are completed in minutes, the rate is
units per minute.
For example, if a worker packs 4 packages every 3 minutes, then the worker's rate is
packages per minute.
Combined Rates
When several people or machines work at the same time, their individual rates can be added.
If their rates are
then their combined rate is
For example, suppose three workers pack 4, 3, and 3 packages every 3 minutes.
Their individual rates are
packages per minute.
Their combined rate is
packages per minute.
Amount Completed
If work is done at a constant rate for units of time, then
Therefore, if the rate changes during the problem, divide the time into intervals and calculate the amount completed during each interval separately.
For example, if the rate is for minutes and then for minutes, the total amount completed is
Workers and can produce 5 and 7 items every 4 minutes, respectively. They begin working together. After minutes, worker , who produces 3 items every 2 minutes, joins them.
If the three workers produce 120 items in 30 minutes, find .
The rates of and are
items per minute.
Thus, their combined rate is
When joins, the combined rate becomes
Therefore,
Multiplying by 2,
Hence,
So worker joins after 10 minutes.
Three machines produce 8, 10, and 15 items every 4, 5, and 6 minutes, respectively.
They begin operating together. After 12 minutes, the second machine stops. How many items have been produced after a total of 30 minutes?
Their rates are
For the first 12 minutes, the combined rate is
Thus, they produce
items.
For the remaining
minutes, the rate is
They produce
more items.
Therefore, the total is
Workers and produce 4 and 5 items every 3 minutes, respectively.
They start together. After 6 minutes, worker , who produces 2 items per minute, joins them. Nine minutes later, worker leaves.
How many items have been produced 24 minutes after and started?
The individual rates are
During the first 6 minutes, the combined rate is
They produce
During the next 9 minutes, all three work:
They produce
The remaining time is
minutes.
Now only and work, at rate
They produce
Therefore, the total is
Workers , , and working together complete 180 items in 24 minutes.
Worker produces 5 items every 2 minutes, and worker produces 8 items every 4 minutes.
How many items does worker produce every 6 minutes?
The combined rate is
The rates of and are
and
Therefore, 's rate is
items per minute.
In 6 minutes, produces
items.
When different rates are given using different time intervals, first convert them to the same unit of time.
For example,
becomes
This makes rates easy to add and compare.
For problems where someone joins or leaves partway through, a timeline is often useful for identifying the different rate intervals.
Do not add statements such as
and
directly. Convert them to the same unit rate first.
The same idea applies to many different situations:
The quantities are different, but the underlying structure is the same: