Last week, I focused on rate problems involving speed (if you're looking to the answer to last week's challenge question, scroll down to the end of this post!). This week, I'm going to shift to work-rate problems, which some students find even more challenging. You know these problems... the ones that say something ridiculous like:
A cyborg can pet a kitten 150 times in 2 minutes. A ninja pets a kitten at half the rate of the cyborg. 3 zombies can pet 140 kittens in 5 hours. What is the difference in the amount of time it would take 30 cyborgs to pet 2000 kittens and the amount of time it would take 40 ninjas and 40 zombies working together to pet the same number of kittens? (Assume the following: 1) All rates are constant. 2) All kittens are not only cute but identical. 3) No ninjas or zombies do battle with each other. 4) No kittens are devoured by the zombies.)
You see a problem like that, and often you're ready to throw in the towel. One particularly aggravating thing is the concept of a combined work rate. You're asked not about the individual ninja and zombie rates, but the amount of time involved if both groups work together.
Let's see how this applies to an actual GMAT question:
It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective constant rates, to complete the order?
A) 7/12 B) 1 1/2 C) 1 5/7 D) 3 1/2 E) 7
So how to deal with this?
Well, it's important to recognize that since the two machines are working together, you can look at a discrete block of time to determine what fraction of the job is done after that amount of time. In this case, let's see what happens after one hour. We know that the first machine takes 4 hours to complete the job. That means that in 1 hour, the machine completes 1/4 of the job. Since the second machine takes 3 hours to complete the job, it will complete 1/3 of the job in 1 hour.
So in the same one-hour span, the two machines work together and complete 1/3 + 1/4 = 7/12 of the job.
But we're not interested in 7/12 of the job; we want to know the amount of time for the whole job. No problem - we can simply use proportions to our advantage:
(7/12 of the job) / (1 hour) = (1 job) / (x hrs)
We must multiply 7/12 by 12/7 to get to 1 job, therefore we must multiply 1 hour by 12/7 to the time in question. So the amount of time we're looking for (and the final answer) is 12/7 = 1 5/7 hours.
For those of you who like formulas, you'll notice that we can come up with a generic equation for combined time. Just repeat what we did before with variables:
Let's say the first machine takes x hours to complete the job. That means that in 1 hour, the machine completes 1/x of the job. Let's say the second machine takes y hours to complete the job. That means it will complete 1/y of the job in 1 hour.
So in the same one-hour span, the two machines work together and complete 1/x + 1/y of the job. Create a common denominator for the left side, and we get: 1/x + 1/y = y/(xy) + x/(xy) = (x+y)/(xy) of the job.
But we want the whole job, so we just have to take the reciprocal (as we did with 7/12 and 12/7). So if (x+y)/(xy) of the job gets done in 1 hour, then the entire job gets done in (xy)/(x+y) hours.
So to sum up, if the first machine takes x hours to complete the job, and the second takes y hours, then the two machines working simultaneously would take (xy)/(x+y) hours to complete the job.
Keep in mind that this is for time only! You can't use this quantity for rate or amount of work. But it can prove a handy shortcut on a more difficult question, such as the following Official GMAT problem (hint hint!).
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce (5/4)w widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?
A) 4 B) 6 C) 8 D) 10 E) 12
Give it a crack, and post your step-by-step solution in the comments!
Epilogue: Last week, I gave you guys the following problem and asked you guys to find the clever shortcut:
Car X and Car Y traveled the same 80-mile route. If Car X took 2 hours and Car Y traveled at an average speed the was 50 percent faster than the average speed of Car X, how many hours did it take Car Y to travel the route?
(A) 2/3 (B) 1 (C) 1 1/3 (D) 1 3/5 (E) 3
Did you guys find it? It turns out that the 80-mile distance is irrelevant; you don't need it at all to do the problem. Remember, since Distance = Rate * Time, and since the two cars are going the same distance, then:
(Rate of Car X) * (Time for Car X) = (Rate of Car Y) * (Time for Car Y)
Rate of Car X) * (2 hours) = (1.5 * Rate of Car X) * (Time for Car Y) <---multiplying by 1.5, since the speed of Car Y is 50% faster
2 hours = 1.5 * Time for Car Y
Time for Car Y = 2/1.5 = 1 1/3 hours
Another way to think about it is to simply realize that since the distance is constant, rate and time are inversely proportional. So if we multiply Car X's rate by 1.5 to get Car Y's rate, that means we divide Car X's time by 1.5 to Car Y's time.
This post was written by Rich Zwelling. For more GMAT prep advice, check out the Knewton GMAT blog.