5. Rates and work
Rates add, times never. Partial work done separately, drains are negative rates.
Core ideas
- Rates add; times never do. Convert every worker to jobs per hour first.
- Time together = 1 / (sum of rates); for two workers, ab/(a+b) where a and b are their solo times.
- Partial work: multiply rate by time worked to get the fraction done, then ask who finishes the rest.
- A drain is a negative rate: subtract it from the filling rates.
Worked example 1
Machine A makes a batch in 4 hours and machine B in 6 hours. A works alone for 1 hour, then B joins. How long do they work together to finish?
Show solution
Rates: A = 1/4 batch per hour, B = 1/6. In the first hour A completes 1/4, leaving 3/4. Combined rate = 1/4 + 1/6 = 5/12 batch per hour. Remaining time = (3/4) / (5/12) = (3/4) * (12/5) = 9/5 = 1.8 hours, or 1 hour 48 minutes together.
Worked example 2
A pipe fills a tank in 3 hours, a second pipe fills it in 5 hours, and a drain empties a full tank in 6 hours. With all three open, how long to fill the empty tank?
Show solution
Net rate = 1/3 + 1/5 - 1/6. Use 30 as the common denominator: 10/30 + 6/30 - 5/30 = 11/30 tank per hour. Time = 30/11 hours, about 2.73 hours (2 hours 44 minutes). The drain is just a rate with a minus sign; nothing else changes.
Practice set
Question 1
A machine produces 240 units in 8 hours at a constant rate. At this rate, how many units does it produce in 5 hours?
Find the per-hour rate first.
Rate = 240/8 = 30 units per hour. In 5 hours: 30*5 = 150 units. Choice D (160) comes from misreading the proportion as 240*(2/3).
Question 2
Working together at their constant rates, pipes A and B fill a tank in 4 hours. Pipe A alone fills the tank in 6 hours. How many hours does pipe B alone need to fill the tank?
Subtract A's rate from the combined rate.
Combined rate is 1/4 and A's rate is 1/6, so B's rate = 1/4 - 1/6 = 3/12 - 2/12 = 1/12 tank per hour. B alone takes 12 hours. Choice A (2) is the trap of subtracting the times 6 - 4.
Question 3
Pipe A fills a tank in 3 hours, pipe B fills it in 6 hours, and drain C empties the full tank in 4 hours. If the tank starts empty and A, B, and C are all open, how many hours does it take to fill the tank?
Sum the two filling rates and subtract the drain's rate.
Net rate = 1/3 + 1/6 - 1/4 = 4/12 + 2/12 - 3/12 = 3/12 = 1/4 tank per hour, so the tank fills in 4 hours. Choice A (2) comes from ignoring the drain (1/3 + 1/6 = 1/2).
Question 4
Working together at their constant rates, A and B can complete a project in 12 days. They work together for 4 days, then A leaves and B finishes the remaining work alone in 24 days. How many days would A need to complete the entire project alone?
Use the 4 days together to find the fraction left, get B's rate from the solo stretch, then back out A's rate.
In 4 days at rate 1/12, they finish 4/12 = 1/3, leaving 2/3. B does 2/3 in 24 days, so B's rate = (2/3)/24 = 1/36 project per day. A's rate = 1/12 - 1/36 = 3/36 - 1/36 = 2/36 = 1/18, so A alone takes 18 days. Choice E (36) is B's solo time.