Anupa-
Great question!
The RTD chart is simply a way of organizing your work, using the concept that Rate x Time = Distance. Here's an extremely simple example:
If I am driving at 45 miles per hour, how long would it take me to drive 405 miles?
I could plug into my chart as follows:
R (mi/hr)_________T (seconds)_______=______D (miles)
45________________ T _____________ =______405
(I often include units in my charts to ensure that I am organizing things correctly. And I know that 45 is my rate because I see the word PER in there. Also, I know that 405 is my distance because MILES is a measure of distance)
So if I set the chart up correctly, I can just divide both sides by 45 to solve for T = 9 hours.
This chart starts to pay off on more difficult, GMAT-like examples. Here's one (OG13, PS#86, pg 164):
Machine A produces bolts at a uniform rate of 120 every 40 seconds, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?
________________________R (bolts/ sec)___x____T (sec)______=______W (bolts)
Machine A________________A___________________40 ______________ 120
Machine B ________________B___________________20_______________100
Note that this is RTD in a different form - instead of distance, I am calculating work. But since Rate x Time = Work, the same concept applies (as does the chart)! Since I know that A x 40s = 120 bolts, A must be 3 bolts/second. Since I know that B x 20s = 100 bolts, I know that B must be 5 bolts per second. When two machines are working together, then I add their rates together to get their combined rate-just like if I make 2 paper airplanes per second, and my friend Tom makes 3 paper airplanes per second, then when Tom and I work together, we have a combined rate of 5 paper airplanes per second. Pretty impressive folding times.
In this GMAT case, both machines working together would thus produce 8 bolts / second. The final step is finding out how long it will take the two machines working at that rate to create 200 bolts. I can just plug that scenario into my chart (bottom row) and solve:
_________________________R (bolts/sec)______x______T (sec)____=______W (bolts)
Machine A_____________________3____________________40________________120
Machine B _____________________5____________________20________________100
A and B working together_______8____________________T_________________200
I know 8T = 200, so T = 25 sec (Answer Choice B).
So RTD and RTW are closely related, as are other problems involving rates. An interesting twist on these types of problems is exemplified in the difficult one below (PS #151 from the OG13, pg 173):
At a loading dock, each worker on the night crew loaded ¾ as many boxes as each worker on the day crew. If the night crew has 4/5 as many workers as the day crew, what fraction of all the boxes loaded by the two crews did the day crew load?
In this problem, I am asked to analyze the number of workers, the number of boxes PER worker (this is a rate!), and the total number of boxes. Multiplying my rate by the number of workers will yield my total number of boxes. Thus, I can use the same concept as my RTD chart and organize my work in the exact same manner.
I would set up my chart as follows and plug in according to the ratios given (these are not the only values that work for this particular example, but they are probably the most simple ones):
________Rate (boxes / worker)______# of workers_________# of Boxes
Day__________4_________________________5___________________20
Night_________3_________________________4__________________12
Total ___(from adding boxes loaded by day & night crews)____32
The fraction I am asked for is (Boxes loaded by the day crew) / (Total number of boxes), or 20 / 32, which simplifies to 5 / 8 (Answer choice E).
This is an extremely useful type of chart to master!
I hope that helps.
-Dan Lerman
Manhattan Prep
