Work Problems
Combined worker - one object has an unknown time
We can represent the rate at which it can complete the job as 1/t. We can solve for t to determine the object's rate or the time it takes the object to complete a job by itself.
Adding combined rates
A GMAT question also may provide the rate for two or more objects, in which case we're expected to compute the "combined rate" of those objects. We compute this combined rate by adding together the individual rates of the objects. 1/x + 1/y = combined rate. We can also add and subtract combined rates.
Relative Rate Work problems - The rate of one worker is expressed as a multiple of the rate of another worker
If object 1 is x times as fast as object 2, we let object 2's rate be the variable r, objects 1's rate would be xr.
Relative Rate Work problems - The rate of one worker is slower or faster than the rate of another worker
If object 1 takes x minutes longer than object 2 to complete a job and we let object 2's time be represented by the variable t minutes, object 1's time would be (t+x) minutes. Object 1's rate would be 1/(T+x) and object 2's rate would be 1/t
Relative Rate Work problems - One worker can complete a job in some percent greater or less than the time it takes another worker to do the same job
If object 1 takes x percent fewer minutes than object 2 to complete a job and we let object 2's work time be represented by variable t minutes, object 1's work time would be t(100-x/100) the rate would then be 1 over this.
Combined worker - one object stops before completion
Let the work time for the object that stops first be represented by x and the work time for the object that finished the job alone be represented by (x+y), with y representing the additional time needed to complete the job x/6 + (x+y/8) = 1 and usually solve for y
The Proportion Method
Start by defining the rate of the five workers. 1/6. Next we can use the following proportion to determine the rate of two workers: x workers/combined rate of workers = y workers/combined rate of y workers 5/1/6 = 2/n n = 1/15
Single Worker Problems
Test basic understanding and ask us to either find the rate at which an object works, the time the object spends working, or the amount of work performed by an object. i.e. Machine A takes two hours to assemble 40 widgets. If Machine B is fifty percent faster than Machine A, how many hours would it take for Machine B to assemble 300 widgets? Rate of Machine A = 40/2 = 20 widgets an hour Rate of Machine B = 20 * 3/2 = 30 widgets an hour Time of b = 300/30 = 10
Determining an objects' work rate
The rate at which an object is performing a task or job can be expressed as work/time. This can sometimes show up as a fraction of a job in a specific amount of time. 1/4 job completed in 2 hours is equal to 1 job/8hours
Change in workers problems
These test on what happens to the time it will take a group of workers to complete a task when a certain number of workers are added to or removed from a group. A key component of these problems is determining the overall rate after workers have been added or removed. Two methods can be used to determine this new rate: 1. Defining the rate of one worker 2. The proportion method
Defining the rate of one worker
We can see that one job is performed by five workers in 6 days. the rate of the five is 1/6. 1/6 of the job is completed in a day. Next, we determine the rate of one worker by dividing the rate of the five workers by five, and we have 1 job/30 days. Finally, to determine the combined rate of the two remaining workers, we multiply the rate of one worker by 2. so we have 1 job/15 days.
Combined Worker Problems
When two or more objects are working together. Work total = work of object 1 + work of object 2 Remember that work = time*rate
Work equation
Work = rate x time Rate = work/time Time = work/rate All of the units of measurement in this equation must match
Opposing Worker problems
sometimes a problem will present people or objects that are working in opposition. In these problems, remember to subtract the work done by one object from the work done by the other object. i.e. Working at a constant rate, with the drain closed, a particular faucet can fill an empty tank to the top in 20 minutes. Working at a constant rate, with the faucet not turned on, a drain can empty the same full take in 36 minutes. If the faucet began filling the empty tank with the drain open, how long would it take to fill the tank to the top? t/20 + t/36 = 1
Combined worker - Percent of a job done and fraction of a job done
Say me and a friend built a bookcase. I worked at a constant rate of 1 bookcase/4 hours and my friend worked at a rate of 1 bookcase/8 hours. If we both worked for the same amount of time, what fraction of the bookcase did I build? First, we need to determine the total work that is performed. since the time is the same, it can be represented by t. 1t/4 + 1t/8 = 3t/8. The fraction of the total job that I did is (1t/4)/(3t/8). I performed 3/8 of the job. **It's important to see that in this problem, it is not necessary to determine the value of t before determining the percent or faction of the job I completed.
