Word Problems
(Work-Rate)
Let’s discuss hard difficulty level scenarios of Work-Rate problem at this stage:
A wider pipe can fill 2.5 times faster than a narrow pipe. Working together, both pipes can fill the pool in 2.5 hours. How long will it take to fill the pool if wider pipe work alone?
Solution:
Suppose if it were written that wider pipe can fill 2 times faster than narrow pipe, what you think about this version. Obviously, wider pipe is working at twice rate than narrow one. Similarly, the 2.5 times is actually 2.5 times the rate of narrow pipe. That’s it, remaining solution is as simple as we did in previous working.
So, let’s say narrow pipe rate is 1⁄A
⇒ Rate of wider pipe = 2.5 × (1⁄A) = (5⁄2) × (1⁄A)
⇒ Rate of wider pipe = 5⁄2A
Remember, don’t make mistake here by supposing A as time required by narrow pipe. Always take rate into consideration rather than time in work-rate problem.
Now do same thing as we did before.
{(5⁄2A) + (1⁄A)} × 2.5 = 1
⇒ {(5⁄2A) + (1⁄A)} × (5⁄2) = 1
⇒ {(5 + 2)⁄2A} = 2⁄5
⇒ 35 = 4A
⇒ A = 8.75 hours Answer
So you’ve learned here always you should take individual rates by reversing the individual time. (i.e if time required is 4 hours to do a job, then the rate would be simply 1⁄4). That’s the step you have to remember while doing any work-rate problem of working together scenario.
Now, let’s do a more complex scenario to make you feel more comfortable in this topic.
Working simultaneously Machine M can produce a units in 3⁄4 of the time it takes machine N to produce the same amount of units. Machine N can produce a units in 2⁄3 the time it takes machine O to produce that amount of units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N?
Solution:
This question is not as hard as it seems. Just you need to be well control on your nerves during exam day, while solving questions like this one. Trust me it’s not very hard. You can do this. 🙂
Let’s solve this.
As this is of working together scenario, So we have to find the individual rates of each machines. Note than here work no more 1, rather it is a now.
Machine Work Time
M a tm = 3tn⁄4
N a tn = 2to⁄3
O a to
If any of you not understood what happen in above expressions, you are most welcome to ask me on whatsApp at (+923214711387). As I want to improve your intellect bit by bit, so that’s why I’ve started things little faster slowly. So here, majority of students would learn this comfortably.
From the above three expressions, we can say that
At this stage, we have individual time required by each Machine to make a units. And we convert each time in same variable, i.e to. We should reverse each time and multiply by work to get rates of each Machine. In other words, we need to divide time by work to get rates of each Machines as below:
Rate of M = 2a⁄to
Rate of N = 3a⁄2to
Rate of O = a⁄to
Now, by using working together scenario and using work-rate basic equation, we’ll get:
{(2a⁄to) + 3a⁄2to + a⁄to)} × Time = a
⇒ a{(4 + 3 + 2)⁄2to} × T = a (By taking a common)
⇒ {9⁄2to} × T = 1
⇒ T = 2to⁄9
We are required to find the part of total work done by Machine N, i.e we are required to find Work done by N⁄Total Work,
As work done by N = rate of N × Time = 3a⁄2to × 2to⁄9 (As T = 2to⁄9 )
⇒ Work done by N = a⁄3
Therefore, Work of N as fraction of total work = a⁄3 ÷ a = a⁄3 × 1⁄a
⇒ Work of N as fraction of total work = 1⁄3 Answer
Let’s discuss a scenario in work-rate, that seems very hard but actually it’s very easy, normally I call it as ‘halwa question’ term used in Urdu for question whose answer is in fingertips.
At their respective rates, pump A, B, and C can fulfill an empty tank, or pump-out the full tank in 2, 3, and 6 hours. If A and B are used to pump-out water from the half-full tank, while C is used to fill water into the tank, in how many hours, the tank will be empty?
Solution:
Let’s remain to the point and shorten our calculation such that average students can easily understand at ease.
Rate of pump A = 1⁄2
Rate of pump B = 1⁄3
Rate of pump C = 1⁄6
While remaining work = 1⁄2 (AS tank is half filled, that is required to make empty)
Now, C pump the water into the tank, while two powerful pumps A and B pump the water out of the tank. So all in all, the net rate would be:
Net rate of water pump-out = 1⁄2 + 1⁄3 – 1⁄6 = (3 + 2 – 1)⁄6 = 4⁄6 = 2⁄3
Now, by using basic equation, we can find time required by the three pumps working together to make the tank empty.
⇒ (2⁄3) × Time = 1⁄2 (As only half the tank is required to empty)
⇒ T = 1⁄2 × 3⁄2 = 3⁄4 hours =
⇒ T = 45 minutes Answer
Finally, let’s do the hardest possible scenario that could be tested in GMAT or GRE exam.
If 4 men or 7 boys (each with identical rates) can finish a task in 29 days then how long would it take 12 men and 8 boys to finish the same task?
Solution:
This question is the combination of the two basic scenarios that we have discussed in first part of ‘work-rate’ topic. One the change in same variable, and other working together scenario.
First, we need to find the rate of 12 men as well as rate of 8 boys, by using first scenario method; and then we’ll find time required by these 12 men and 8 boys, by using working together scenario method. So,
As there’s inverse relationship, so we’ll do horizontal multiplication rather than cross multiplication,
⇒ 4 × 29 = 12x
⇒ x = 29⁄3
Therefore,
Rate of 12 Men = Rm = 3⁄29 (As rate would be reversed)
Here again there’s inverse relationship, so we’ll do horizontal multiplication rather than cross multiplication,
⇒ 7 × 29 = 8y
⇒ y = (7 × 29)⁄8
Important Note: For GMAT students, do not multiply 7 and 29, because in later on, perhaps you’ll need to cancel down some of the prime number (i.e 2, 3, 5, 7, 11, …). If you’ll multiply 7 and 29 here, later on you’ll need to break it down again, so it’ll be waste of your time twice. So always try to make expression as it is till you have the answer in hand. If you realize it is the answer, then you may multiply. For GMAT, it is very important to save time. Further important points you’ll learn in Arithmetic topic later on.
Therefore,
Rate of 8 Boys = Rb = 8⁄(7 × 29) (As rate would be reversed)
Now we’re required to find the time taken by 12 Men and 8 Boys i.e working together.
⇒ (Rm + Rb) × Time = 1
⇒ {3⁄29 + 8⁄(7 × 29)} × T = 1
⇒ {(21 + 8)⁄(7 × 29)} × T = 1 (As L.C.M of 7 and 29 is simply multiplication of the these)
⇒ {29⁄(7 × 29)} × T = 1
⇒ {1⁄7} × T = 1
You see that why I advised you before for GMAT students. If you multiplied 7 and 29 before, this would span lot more time.
⇒ T = 7 days Answer