📘 SAT MATH: WORK RATE PROBLEMS
𝗛𝗼𝘄 𝘁𝗼 𝗦𝗼𝗹𝘃𝗲 𝗖𝗼𝗺𝗯𝗶𝗻𝗲𝗱 𝗪𝗼𝗿𝗸, 𝗣𝗶𝗽𝗲, 𝗠𝗮𝗰𝗵𝗶𝗻𝗲 𝗮𝗻𝗱 𝗥𝗮𝘁𝗲 𝗣𝗿𝗼𝗯𝗹𝗲𝗺𝘀
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Work rate problems can look like long word problems, but the mathematics underneath is usually built from one simple idea:
𝗥𝗮𝘁𝗲 × 𝗧𝗶𝗺𝗲 = 𝗪𝗼𝗿𝗸
The difficulty comes from translating the wording.
A person may finish a task in a certain number of hours.
A machine may produce a certain number of parts every minute.
A pipe may fill a tank while another pipe drains it.
Two workers may begin at different times.
One worker may leave before the job is finished.
These situations look different, but they can all be handled by thinking about how much work is completed during one unit of time.
This guide builds the method from the beginning and then moves into the harder forms that can appear in SAT-style rate and word problems.
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1️⃣ 𝗧𝗛𝗘 𝗠𝗢𝗦𝗧 𝗜𝗠𝗣𝗢𝗥𝗧𝗔𝗡𝗧 𝗜𝗗𝗘𝗔
Suppose Maya can complete a job in 8 hours.
If the entire job is called 1 job, then in one hour Maya completes:
1/8
of the job.
So her rate is:
1/8 job per hour
Now suppose Leo can complete the same job in 12 hours.
His rate is:
1/12 job per hour
If they work at the same time, the amount completed during one hour is:
1/8 + 1/12
= 3/24 + 2/24
= 5/24
So together they complete:
5/24 of the job per hour
The time needed for one complete job is therefore:
1 ÷ 5/24
= 24/5
= 4.8 hours
So:
Answer: 4.8 hours
This is the central idea behind almost every SAT work rate problem:
𝗔𝗱𝗱 𝗿𝗮𝘁𝗲𝘀, 𝗻𝗼𝘁 𝘁𝗶𝗺𝗲𝘀.
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2️⃣ 𝗪𝗛𝗔𝗧 𝗜𝗦 𝗔 𝗪𝗢𝗥𝗞 𝗥𝗔𝗧𝗘 𝗣𝗥𝗢𝗕𝗟𝗘𝗠?
A work rate problem tells you how quickly something gets completed.
The “something” could be:
• a construction job
• a repair
• a cleaning task
• a batch of products
• a number of pages
• a tank being filled
• a tank being emptied
• a computer process
• a production order
The people or machines are simply different ways of describing rates.
For example:
A worker finishes a task in 6 hours.
This means:
Rate = 1/6 job per hour
A machine produces 80 parts per hour.
This means:
Rate = 80 parts per hour
A pump removes 15 liters per minute.
This is a rate in the opposite direction:
−15 liters per minute
The numbers look different, but the basic concept is identical.
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3️⃣ 𝗧𝗛𝗘 𝗕𝗔𝗦𝗜𝗖 𝗙𝗢𝗥𝗠𝗨𝗟𝗔
The fundamental relationship is:
Work = Rate × Time
From this:
Rate = Work ÷ Time
and:
Time = Work ÷ Rate
When a problem describes one complete job, it is usually convenient to let:
Total work = 1
Then a worker who takes t hours has a rate of:
1/t
This reciprocal relationship is why fractions appear so often in SAT combined-work problems.
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4️⃣ 𝗧𝗪𝗢 𝗣𝗘𝗢𝗣𝗟𝗘 𝗪𝗢𝗥𝗞𝗜𝗡𝗚 𝗧𝗢𝗚𝗘𝗧𝗛𝗘𝗥
Suppose:
Worker A finishes a job in a hours.
Worker B finishes the same job in b hours.
Their rates are:
1/a
and
1/b
Together:
1/a + 1/b
If T is the time they need together:
1/T = 1/a + 1/b
This can also be simplified to:
T = (a x b)/(a + b)
This is one of the most useful shortcuts for two-worker SAT problems.
Example
One technician completes a repair in 7 hours.
Another technician completes it in 21 hours.
Together:
T = (7 x 21)/(7 + 21)
T = 147/28
T = 21/4
T = 5.25 hours
So:
Answer: 5.25 hours
or:
5 hours 15 minutes
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5️⃣ 𝗪𝗛𝗬 𝗗𝗢𝗡'𝗧 𝗪𝗘 𝗔𝗗𝗗 𝗧𝗛𝗘 𝗧𝗜𝗠𝗘𝗦?
This is one of the most common questions about work rate problems.
Suppose one person takes 10 hours and another takes 15 hours.
Adding the times gives:
10 + 15 = 25
But 25 hours has no meaning as their combined completion time.
The first person completes:
1/10
of the job each hour.
The second completes:
1/15
each hour.
Together:
1/10 + 1/15
= 3/30 + 2/30
= 5/30
= 1/6
So they complete the job at a rate of:
1/6 job per hour
Therefore:
Time = 6 hours
The reason is simple:
When two people work simultaneously, their work completed per hour is combined.
Their individual completion times are not.
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6️⃣ 𝗔 𝗙𝗢𝗨𝗥-𝗦𝗧𝗘𝗣 𝗠𝗘𝗧𝗛𝗢𝗗
When you see a SAT work rate question, use this sequence.
Step 1
Decide what the complete job represents.
Usually:
1 job
Step 2
Convert completion times into rates.
If someone takes 9 hours:
Rate = 1/9
Step 3
Combine the rates.
Workers helping:
Add
Drains or leaks:
Subtract
Step 4
Use:
Time = Work ÷ Rate
Then check the units.
This method works for two workers, three workers, machines, pipes and many harder variations.
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7️⃣ 𝗧𝗛𝗥𝗘𝗘 𝗪𝗢𝗥𝗞𝗘𝗥𝗦 𝗧𝗢𝗚𝗘𝗧𝗛𝗘𝗥
If three workers independently require:
a hours, b hours and c hours,
their combined rate is:
1/a + 1/b + 1/c
Therefore:
1/T = 1/a + 1/b + 1/c
Example
Three machines can finish an order individually in:
4 hours, 8 hours and 16 hours.
Their combined rate is:
1/4 + 1/8 + 1/16
= 4/16 + 2/16 + 1/16
= 7/16
Therefore:
T = 1 ÷ 7/16
= 16/7
hours.
Answer: 16/7 hours
or approximately:
2 hours 17 minutes
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8️⃣ 𝗠𝗔𝗖𝗛𝗜𝗡𝗘 𝗣𝗥𝗢𝗗𝗨𝗖𝗧𝗜𝗢𝗡 𝗣𝗥𝗢𝗕𝗟𝗘𝗠𝗦
Some SAT rate questions give the rate directly instead of giving the completion time.
Suppose:
Machine A produces 48 units per hour.
Machine B produces 32 units per hour.
Together:
48 + 32 = 80 units per hour.
If they need to produce 560 units:
Time = 560 ÷ 80
= 7 hours.
Answer: 7 hours
Notice that no reciprocal calculation was necessary.
The rates were already given.
That leads to an important rule:
𝗜𝗳 𝘁𝗵𝗲 𝗽𝗿𝗼𝗯𝗹𝗲𝗺 𝗴𝗶𝘃𝗲𝘀 𝗿𝗮𝘁𝗲𝘀 𝗱𝗶𝗿𝗲𝗰𝘁𝗹𝘆, 𝘂𝘀𝗲 𝘁𝗵𝗼𝘀𝗲 𝗿𝗮𝘁𝗲𝘀.
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9️⃣ 𝗣𝗔𝗥𝗧𝗜𝗔𝗟 𝗪𝗢𝗥𝗞
A worker may complete only part of a job before something changes.
The formula is:
Work completed = Rate x Time
Example
A machine completes 1/9 of an order every hour.
How much of the order will it complete in 5 hours?
Work:
5 x 1/9
= 5/9
So:
Answer: 5/9 of the order
The unfinished portion is:
1 − 5/9
= 4/9
This becomes especially important when another worker joins later.
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🔟 𝗪𝗛𝗘𝗡 𝗔 𝗦𝗘𝗖𝗢𝗡𝗗 𝗪𝗢𝗥𝗞𝗘𝗥 𝗝𝗢𝗜𝗡𝗦
These questions should be separated into time periods.
Example
A worker can finish a job in 12 hours.
A second worker can finish it in 8 hours.
The first worker works alone for 3 hours. Then the second worker joins.
How long do they work together after that?
First worker's rate:
1/12
Work completed in 3 hours:
3 x 1/12
= 1/4
Remaining:
1 − 1/4
= 3/4
Combined rate:
1/12 + 1/8
= 2/24 + 3/24
= 5/24
Time for the remaining 3/4:
(3/4) ÷ (5/24)
= 3/4 x 24/5
= 18/5
= 3.6 hours
Answer: 3.6 hours
If the question asks for total time from the beginning:
3 + 3.6 = 6.6 hours.
Always check exactly what the question asks.
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1️⃣1️⃣ 𝗪𝗛𝗘𝗡 𝗢𝗡𝗘 𝗪𝗢𝗥𝗞𝗘𝗥 𝗟𝗘𝗔𝗩𝗘𝗦
This is the reverse situation.
Example
Worker A can finish a task in 9 hours.
Worker B can finish it in 6 hours.
They work together for 2 hours. Then A leaves.
How much longer does B need?
Combined rate:
1/9 + 1/6
= 2/18 + 3/18
= 5/18
Work completed in 2 hours:
2 x 5/18
= 10/18
= 5/9
Remaining:
1 − 5/9
= 4/9
B's rate:
1/6
Time required:
(4/9) ÷ (1/6)
= 4/9 x 6
= 8/3
hours.
Answer: 8/3 hours
or:
2 hours 40 minutes
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1️⃣2️⃣ 𝗣𝗜𝗣𝗘 𝗔𝗡𝗗 𝗧𝗔𝗡𝗞 𝗣𝗥𝗢𝗕𝗟𝗘𝗠𝗦
A filling pipe is simply a positive rate.
A drain is a negative rate.
Suppose one pipe fills a tank in 10 hours.
Its rate is:
1/10
Another pipe empties the tank in 15 hours.
Its rate is:
−1/15
With both operating:
1/10 − 1/15
= 3/30 − 2/30
= 1/30
So the net rate is:
1/30 tank per hour
Therefore the tank takes:
30 hours
to fill.
This is exactly the same rate concept used for workers.
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1️⃣3️⃣ 𝗗𝗜𝗥𝗘𝗖𝗧 𝗣𝗜𝗣𝗘 𝗥𝗔𝗧𝗘𝗦
Sometimes the problem gives gallons per minute or liters per minute.
Suppose a pump adds:
36 liters/minute
and a leak removes:
9 liters/minute.
Net rate:
36 − 9 = 27 liters/minute.
For a 675-liter tank:
Time = 675 ÷ 27
= 25 minutes.
Answer: 25 minutes
No reciprocal formula is required because both rates were already given directly.
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1️⃣4️⃣ 𝗨𝗡𝗜𝗧𝗦 𝗠𝗨𝗦𝗧 𝗠𝗔𝗧𝗖𝗛
A very common mistake is combining rates with different time units.
Suppose:
Machine A produces 180 items per hour.
Machine B produces 2 items per minute.
You cannot immediately write:
180 + 2
because the time units are different.
Convert 180 items per hour:
180 ÷ 60 = 3 items per minute.
Now:
3 + 2 = 5 items per minute.
If 400 items are needed:
400 ÷ 5 = 80 minutes.
Answer: 80 minutes
Before adding rates, make sure their units agree.
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1️⃣5️⃣ 𝗪𝗢𝗥𝗞𝗘𝗥-𝗗𝗔𝗬 𝗣𝗥𝗢𝗕𝗟𝗘𝗠𝗦
Some problems assume that every worker has the same efficiency.
In that case:
Workers x Days = Constant
Example
5 workers complete a project in 18 days.
How many days would 15 workers require if all workers work at the same rate?
Total worker-days:
5 x 18 = 90
With 15 workers:
90 ÷ 15 = 6
Answer: 6 days
This shortcut should not automatically be used when workers have different efficiencies.
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1️⃣6️⃣ 𝗘𝗙𝗙𝗜𝗖𝗜𝗘𝗡𝗖𝗬 𝗥𝗔𝗧𝗜𝗢𝗦
Suppose A works twice as fast as B.
If B's rate is:
r
then A's rate is:
2r
Together:
r + 2r = 3r
Example
A is twice as efficient as B.
Together they complete a job in 9 hours.
Combined rate:
1/9
Let B's rate be r.
Then:
3r = 1/9
r = 1/27
A's rate:
2/27
Therefore A's individual time is:
27/2
= 13.5 hours.
Answer: 13.5 hours
B's individual time is:
27 hours.
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1️⃣7️⃣ 𝗙𝗜𝗡𝗗𝗜𝗡𝗚 𝗔 𝗠𝗜𝗦𝗦𝗜𝗡𝗚 𝗪𝗢𝗥𝗞𝗘𝗥
Suppose two workers together finish in 8 hours.
Worker A alone finishes in 12 hours.
Find Worker B's individual time.
Combined rate:
1/8
A's rate:
1/12
Therefore B's rate:
1/8 − 1/12
= 3/24 − 2/24
= 1/24
So B alone takes:
24 hours
This type of question is often easier than it first appears.
You already know the total rate.
Simply remove the known worker's rate.
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1️⃣8️⃣ 𝗛𝗔𝗥𝗗𝗘𝗥 𝗪𝗢𝗥𝗞 𝗥𝗔𝗧𝗘 𝗣𝗥𝗢𝗕𝗟𝗘𝗠𝗦
Some questions hide the individual times behind a variable.
Example
Worker A takes 5 hours fewer than Worker B.
Together they complete a job in 6 hours.
Find the time Worker A takes alone.
Let A's time be:
t
Then B's time is:
t + 5
Their combined rate is:
1/6
Therefore:
1/t + 1/(t + 5) = 1/6
Multiply by:
6t(t + 5)
This gives:
6(t + 5) + 6t = t(t + 5)
6t + 30 + 6t = t² + 5t
12t + 30 = t² + 5t
t² − 7t − 30 = 0
Factor:
(t − 10)(t + 3) = 0
Therefore:
t = 10
or:
t = −3
A time cannot be negative.
So:
A takes 10 hours.
B takes:
10 + 5 = 15 hours.
Check:
1/10 + 1/15
= 3/30 + 2/30
= 5/30
= 1/6
Correct.
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1️⃣9️⃣ 𝗧𝗛𝗘 𝗟𝗖𝗠 𝗠𝗘𝗧𝗛𝗢𝗗: AVOIDING FRACTIONS
Sometimes fractions make a problem look harder than it is.
You can choose a convenient total amount of work.
Suppose:
Machine A completes a job in 5 hours.
Machine B completes it in 10 hours.
Choose:
Total work = 10 units
Then A produces:
10 ÷ 5 = 2 units/hour.
B produces:
10 ÷ 10 = 1 unit/hour.
Together:
2 + 1 = 3 units/hour.
Time:
10 ÷ 3
= 10/3 hours.
Answer: 3⅓ hours
This method is particularly useful when the individual completion times have a convenient common multiple.
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2️⃣0️⃣ 𝗦𝗔𝗡𝗜𝗧𝗬 𝗖𝗛𝗘𝗖𝗞𝗦
After calculating your answer, stop for a few seconds and ask whether it makes sense.
Two workers
The combined time should be less than the faster worker's individual time.
If one worker takes 5 hours and another takes 8 hours, their combined time must be less than 5 hours.
A worker joins
Once another worker joins, the rate should increase.
A drain opens
The net filling rate should decrease.
A worker leaves
The remaining rate should decrease.
More identical workers
More workers should mean less time, assuming the work conditions and efficiency remain the same.
These quick checks can catch many careless errors.
Trap 1: Adding times
Do not add completion times when workers operate simultaneously.
Use rates.
Do not add completion times when workers operate simultaneously.
Use rates.
⚠️ Trap 2: Forgetting the reciprocal
If a worker takes 8 hours, the rate is:
1/8
not 8.
⚠️ Trap 3: Forgetting to convert units
Minutes and hours must be made consistent.
⚠️ Trap 4: Treating a drain as positive
A drain removes water.
Subtract its rate.
⚠️ Trap 5: Answering total time when the question asks for additional time
Read the final sentence carefully.
⚠️ Trap 6: Using worker-days with unequal workers
The worker-days shortcut assumes equal efficiency.
⚠️ Trap 7: Rounding too early
Keep fractions until the final step whenever possible.
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2️⃣2️⃣ 𝗙𝗔𝗦𝗧 𝗦𝗛𝗢𝗥𝗧𝗖𝗨𝗧𝗦
Two workers
T = (a x b)/(a + b)
Three workers
1/T = 1/a + 1/b + 1/c
Direct production
Time = Total units ÷ Units per time
Partial job
Work = Rate x Time
Remaining job
Remaining = 1 − Completed
Equal-efficiency workers
Workers x Days = Constant
Filling and draining
Net rate = Fill rate − Drain rate
These formulas are useful, but the most important skill is knowing which formula fits the situation.
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2️⃣3️⃣ 𝗧𝗘𝗡 𝗙𝗥𝗘𝗦𝗛 𝗣𝗥𝗔𝗖𝗧𝗜𝗖𝗘 𝗤𝗨𝗘𝗦𝗧𝗜𝗢𝗡𝗦
1.
A technician can complete a calibration task in 14 hours. A second technician can complete it in 21 hours. How long will they need if they work together?
2.
A pipe fills a tank in 9 hours, while a drain empties the full tank in 18 hours. If both remain open, how long will the tank take to fill?
3.
A machine produces 72 components per hour. How many hours are needed to produce 540 components?
4.
12 workers can complete a project in 15 days. Assuming all workers have the same efficiency, how many days would 20 workers need?
5.
One worker completes a task in 16 hours and another completes it in 24 hours. They work together for 4 hours. What fraction of the job remains?
6.
Two workers together complete a job in 9 hours. One worker alone takes 15 hours. How long does the other worker take alone?
7.
A pump adds 64 liters per minute to a tank while a leak removes 16 liters per minute. How long will it take to add 960 liters?
8.
Three machines can individually complete an order in 6, 8 and 24 hours. How long will they take together?
9.
Worker A takes t hours to complete a task. Worker B takes t + 6 hours. Together they complete the task in 4 hours. Find t.
10.
A machine completes 3/5 of a production order in 2.4 hours at a constant rate. How long does it take to complete the entire order?
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
2️⃣4️⃣ 𝗔𝗡𝗦𝗪𝗘𝗥𝗦
- 8.4 hours
- 18 hours
- 7.5 hours
- 9 days
- 1/2
- 22.5 hours
- 20 minutes
- 3 hours
- 6 hours
- 4 hours
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