SAT Work Rate Problems: Solving Jobs Done by Individual and Collaborative Effort

A work rate is the fraction of a job completed per unit time. If a worker completes a job in 6 hours, their rate is 1/6 of the job per hour. If another worker completes the same job in 4 hours, their rate is 1/4 per hour. Working together, their combined rate is 1/6+1/4. Finding a common denominator: 2/12+3/12=5/12 of the job per hour. Time to complete the job together: 1 divided by the rate=1/(5/12)=12/5=2.4 hours. The key principle is that when workers combine their efforts on the same job, their rates add; the combined time to complete the job is 1 divided by the combined rate. This insight converts word problems into straightforward algebra.

The general work equation: (rate1)(time)+(rate2)(time)+...=1 (for one complete job). If Worker 1 works for t1 hours at rate r1 and Worker 2 works for t2 hours at rate r2 on the same job, then r1*t1+r2*t2=1. This equation captures any combination of work scenarios.

Example: Worker A completes a task in 8 hours. Worker B completes it in 12 hours. Working together, how long does it take? A's rate: 1/8 per hour. B's rate: 1/12 per hour. Combined rate: 1/8+1/12=3/24+2/24=5/24 per hour. Time to complete together: 1/(5/24)=24/5=4.8 hours. Another example: A working alone for 3 hours, then A and B together for the remaining time, completes the job in 5 total hours. A's rate: 1/8. B's rate: 1/12. Equation: (1/8)*3+(1/8+1/12)*t=1, where t is the time A and B work together. Simplifying: 3/8+(5/24)*t=1. So (5/24)*t=5/8, giving t=5/8*24/5=3 hours. Total time: 3+3=6 hours. When problems describe different time periods or different workers on different parts of a job, break the problem into segments and apply the work equation to each segment, then combine constraints to solve.

A third scenario: Two workers start simultaneously and work for the same time until the job is done. If A's rate is 1/8 and B's rate is 1/12, and they work together for time t: (1/8+1/12)*t=1. Combined rate is 5/24, so t=24/5=4.8 hours (the same as the cooperative calculation). Seeing this problem from multiple angles (explicit time period vs. using combined rate) builds understanding.

Some problems involve multiple jobs or variable effort. Example: Three workers complete different jobs. Worker A builds 5 units per hour. Worker B builds 3 units per hour. Worker C builds 2 units per hour. How many hours to produce 100 units if all work together? Combined rate: 5+3+2=10 units per hour. Time: 100/10=10 hours. This is simpler arithmetic than the fraction-based approach but uses the same principle. Another scenario: A job requires different amounts of work from different workers. Machine X prints 1000 pages per hour. Machine Y prints 800 pages per hour. If X runs for 2 hours and Y runs for 3 hours, how many pages are printed? Pages from X: 1000*2=2000. Pages from Y: 800*3=2400. Total: 4400 pages. The work rate framework adapts to any scenario where effort (quantity per unit time) is given and you need to find total output or time required; the core logic remains: amount done=rate*time, and rates combine when working simultaneously on the same task.

Problems sometimes describe one worker slowing down or speeding up. If A normally works at rate 1/10 (10-hour job) but works slower at 2/3 of normal speed, their new rate is (2/3)*(1/10)=1/15. These adjustments apply directly to the rate formula.

Set up work problems carefully using the rate framework. (1) Identify the rate (fraction of job per unit time, or output per unit time). (2) Write the work equation. (3) Solve for the unknown (time or job size). (4) Check by substituting back. A systematic process prevents errors. Common mistakes: confusing time with rate (if a job takes 6 hours, the rate is 1/6, not 6), adding times instead of rates (rates add, not times), and forgetting to convert to a common unit. A 1-week drill on work rates: Days 1-2, two-worker collaboration problems (find time to complete job together). Days 3-4, find how much is done when workers work for different times. Days 5-6, complex scenarios (three+ workers, variable rates, multiple jobs). Day 7, mixed practice. This builds the intuition to set up these problems automatically.

On test day, mentally convert the problem into a rate statement before writing the equation. "A does the job in 6 hours" means A's rate is 1/6. Once you have rates, addition and the work equation solve almost any scenario. This mental translation is the key skill; the algebra is straightforward once the problem is set up correctly.