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It takes printer A 4 more minutes than printer B to print 40

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Re: a good one [#permalink]

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New post 19 Feb 2014, 00:34
Expert's post
prasannajeet wrote:
Hi Bunuel

I may be crazy but let me understand what is wrong in the following equation and why I am not getting the right answer...
Lets rate of B=T so A=T+4

(1/T+4)+(1/T)=5/24....????? :roll:

Rgds
prasannajeet


If T is the rate how is T + (4 days) = (rate) + (time) = (rate) ???
If T is rate, then 1/T = 1/(rate) = (time), while 5/24 = (rate). How is (1/T+4)+(1/T) = (time) + (time) + (time) = 5/24 = (rate)???
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Re: a good one [#permalink]

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New post 19 Feb 2014, 02:51
Bunuel wrote:
prasannajeet wrote:
Hi Bunuel

I may be crazy but let me understand what is wrong in the following equation and why I am not getting the right answer...
Lets rate of B=T so A=T+4

(1/T+4)+(1/T)=5/24....????? :roll:

Rgds
prasannajeet


If T is the rate how is T + (4 days) = (rate) + (time) = (rate) ???
If T is rate, then 1/T = 1/(rate) = (time), while 5/24 = (rate). How is (1/T+4)+(1/T) = (time) + (time) + (time) = 5/24 = (rate)???


Sorry for typo...
T is the time taken by B then T+4 is time taken by A.
So now the rate would be
40/T+40/T+4=50/6....Still wont get the right answer...

Pls Help.....

Rgds
Prasannajeet
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Re: a good one [#permalink]

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New post 19 Feb 2014, 03:36
Expert's post
prasannajeet wrote:
Bunuel wrote:
prasannajeet wrote:
Hi Bunuel

I may be crazy but let me understand what is wrong in the following equation and why I am not getting the right answer...
Lets rate of B=T so A=T+4

(1/T+4)+(1/T)=5/24....????? :roll:

Rgds
prasannajeet


If T is the rate how is T + (4 days) = (rate) + (time) = (rate) ???
If T is rate, then 1/T = 1/(rate) = (time), while 5/24 = (rate). How is (1/T+4)+(1/T) = (time) + (time) + (time) = 5/24 = (rate)???


Sorry for typo...
T is the time taken by B then T+4 is time taken by A.
So now the rate would be
40/T+40/(T+4)=50/6....Still wont get the right answer...

Pls Help.....

Rgds
Prasannajeet


Solving 40/T+40/(T+4)=50/6 gives T=8 minutes, thus printer A to print 40 pages takes T+4=12 minutes. The same answer as we have here: it-takes-printer-a-4-more-minutes-than-printer-b-to-print-98479.html#p759109

Hope it's clear.
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Re: a good one [#permalink]

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New post 19 Feb 2014, 04:05
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Nothing's wrong here. Your equation: [{1/(T+4)} + {1/T}] = 5/24 is a simplified version of the actual equation: \(6*[{40/(T+4)}+{40/T}]=50\).

Which essentially has come from (what has been explained in above posts); i.e. In 1 minute, A can print 40/T+4 pages and B can print 40/T pages, so together in 1 min they can print 40/(T+4) + 40/T pages. Therefore, in 6 minutes, they can print 6*[{40/(T+4)}+{40/T}], which is given to be 50. When you simplify, you get the equation that you have written.

Pardon the repetitions or my lack of brevity :)

prasannajeet wrote:

Hi Bunuel

I may be crazy but let me understand what is wrong in the following equation and why I am not getting the right answer...
Lets rate of B=T so A=T+4

(1/T+4)+(1/T)=5/24....????? :roll:

Rgds
prasannajeet
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Re: It takes printer A 4 more minutes than printer B to print 40 [#permalink]

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Re: It takes printer A 4 more minutes than printer B to print 40 [#permalink]

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New post 09 Jan 2016, 14:55
VeritasPrepKarishma, please see if you can help me.

At your solution you write 1/a + 1/(a - 8) = 5/48, with only the number 1 at the denominator, very different from the other solutions that usually use the number 40 at the denominator.

Can you please explain?

Regards.

VeritasPrepKarishma wrote:
siddharthmuzumdar wrote:
Got the quadratic soon but took ages after that to solve the rest of the problem. The substitution, I guess, would take up a lot of time. How to solve within 2 mins?


Do not go up to the quadratic to solve it. The substitution will take a long time. Get your basic equation and then substitute. Let me show you what I mean.

Let me make the work same.
We need time taken by printer A to print 80 pages. (Let's say this is 'a' mins)
We know A takes 4 more mins for 40 pages so it will take 8 more minutes for 80 pages.
Together, they print 50 pages in 6 mins so they will print 80 pages in 6*50/80 = 48/5 mins

Now, make your sum of rates equation:

1/a + 1/(a - 8) = 5/48

Now look at the options and substitute here. First check out the straight forward options.
Say, a = 12
1/12 + 1/4 = 4/12 Nope

I will not try 18 and 20 because (18, 10) and (20, 12) doesn't give me 48, the denominator on right hand side.

I will try 24 instead.
1/24 + 1/16 = 5/48 Yes.
Answer is 24.
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Different methods to solve a Work and Time problem [#permalink]

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New post 10 Jan 2016, 02:57
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Hello Everyone,

Today I just happened to come across a work and time problem, which had a 95% difficulty level. I immediately thought of a way to do it and realized two ways were already discussed but not this one. All three would fall into category of standard methods. This prompted me to see if a fourth method could be found, a POE(Process Of Elimination) method. And I realized it was faster and brought down the difficulty level by a bit of thinking.
The point is a bit of thinking can actually give us clues to try to solve the Qs in lesser time and different ways. But its important for that to try out different methods when we are practicing.

Here I will touch upon all 4 methods that can be used to solve the Q.
I would prefer POE if the choices have values spread apart, otherwise any of the remaining three.

Question is:-
It takes printer A 4 more minutes than printer B to print 40 pages. Working together, the two printers can print 50 pages in 6 minutes. How long will it take printer A to print 80 pages?

A. 12
B. 18
C. 20
D. 24
E. 30

1)

POE..


yes, you can use the method of elimination to get to the answer...
what do we do?... we try and find the MINIMUM and MAXIMUM value it can take and then check for the choices..

first step:-


combined both do 50 pages in 6 minutes.
80 pages can be done in 6*80/50=48/5= 9 minutes 36 seconds..

second step:-


If there is a difference of 4 minutes while doing 40 pages, the difference will increase to 8 minutes for 80 pages..
We have one slower,A and one faster, B...
A and B do 80 pages in 9min 36 secs..
Both will do 2times 80 pages in 19 min 12 secs...
2 of As will do 2 times 80 pages in > 19 min 12 secs,A works with a faster machine to clock a time of 19 min 36 secs...
or A will do 1 time 80 pages >19 min 12 secs..


so 19 min 12 secs becomes our min time if the two A and B operate at the same speed..,
but ofcourse it should be much more, since there is a 8 minutes diff between the two....

this eliminates all except 24 and 30..

third step:-


lets find the max it can be ..
the difference between A and B is 8 min..
B is faster than A, so should take less their combined time 19 min 12 secs...
Although A will be lesser than but, in no way, it can go beyond 19 min 12 secs. + 8 min, if we take B as 19 min 12 secs, the max possible (we are taking this,although in reality it will be lesser to check max value of A).
so max time = 19 min 12 secs. + 8 min = 27 min 12 secs..
this eliminates 30 too..
ans 24 min...

2)

Second method


We can work on 50 pages and 6 min as there are two values avail..
there is a difference of 4 mins in 40 pages..
so, there will be a difference of 5 mins in 50 pages..
let the time taken by A be x min, then B will take x-5 min..
their combined one minute work= 50/x + 50/(x-5)...
combined they do 50 pages in 6 min, so they will do 50/6 in one minute..
so \(\frac{50}{x} + \frac{50}{{x-5}} = \frac{50}{6}\)..
removing 50 from both sides \(6x-30 + 6x= x^2-5x\)...
x^2-17x+30 =0.....
x=15 or -2...
it cannot be negative, therefore x=15..
now A does 50 pages in 15 min, so it will do 80 pages in 15*80/50 = 24 min..

3)

Another Standard method


(I am copying from the earlier post, so thanks @Bunuel)
Quote:
Let the time needed to print 40 pages for printer A be \(a\) minutes, so for printer B it would be \(a-4\) minutes.

The rate of A would be \(rate=\frac{job}{time}=\frac{40}{a}\) pages per minute and the rate of B \(rate=\frac{job}{time}=\frac{40}{a-4}\) pages per minute.

Their combined rate would be \(\frac{40}{a}+\frac{40}{a-4}\) pages per minute. Also as "two printers can print 50 pages in 6 minutes" then their combined rate is \(rate=\frac{job}{time}=\frac{50}{6}\), so \(\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}\).

\(\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}\) --> \(\frac{1}{a}+\frac{1}{a-4}=\frac{5}{24}\). At this point we can either try to substitute the values from answer choices or solve quadratic equation. Remember as we are asked to find time needed for printer A to print \(80\) pages, then the answer would be \(2a\) (as \(a\) is the time needed to print \(40\) pages). Answer D works: \(2a=24\) --> \(a=12\) --> \(\frac{1}{12}+\frac{1}{8}=\frac{5}{24}\).


4)

another way in the same thread

..

We need time taken by printer A to print 80 pages. (Let's say this is 'a' mins)
We know A takes 4 more mins for 40 pages so it will take 8 more minutes for 80 pages.
Together, they print 50 pages in 6 mins so they will print 80 pages in 6*50/80 = 48/5 mins

Now, make your sum of rates equation:

1/a + 1/(a - 8) = 5/48

Now look at the options and substitute here. First check out the straight forward options.
Say, a = 12
1/12 + 1/4 = 4/12 Nope

I will not try 18 and 20 because (18, 10) and (20, 12) doesn't give me 48, the denominator on right hand side.

I will try 24 instead.
1/24 + 1/16 = 5/48 Yes.
Answer is 24...

Finer points


1. there can be various methods to solve the Qs, by substitution, by standard algebra, or by POE..
2. we have to see how each fits in, but these have to be practiced in abundance to get a feel of each method.
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Re: It takes printer A 4 more minutes than printer B to print 40 [#permalink]

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New post 10 Jan 2016, 21:29
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mestrec wrote:
VeritasPrepKarishma, please see if you can help me.

At your solution you write 1/a + 1/(a - 8) = 5/48, with only the number 1 at the denominator, very different from the other solutions that usually use the number 40 at the denominator.

Can you please explain?

Regards.

VeritasPrepKarishma wrote:
siddharthmuzumdar wrote:
Got the quadratic soon but took ages after that to solve the rest of the problem. The substitution, I guess, would take up a lot of time. How to solve within 2 mins?


Do not go up to the quadratic to solve it. The substitution will take a long time. Get your basic equation and then substitute. Let me show you what I mean.

Let me make the work same.
We need time taken by printer A to print 80 pages. (Let's say this is 'a' mins)
We know A takes 4 more mins for 40 pages so it will take 8 more minutes for 80 pages.
Together, they print 50 pages in 6 mins so they will print 80 pages in 6*50/80 = 48/5 mins

Now, make your sum of rates equation:

1/a + 1/(a - 8) = 5/48

Now look at the options and substitute here. First check out the straight forward options.
Say, a = 12
1/12 + 1/4 = 4/12 Nope

I will not try 18 and 20 because (18, 10) and (20, 12) doesn't give me 48, the denominator on right hand side.

I will try 24 instead.
1/24 + 1/16 = 5/48 Yes.
Answer is 24.


In my solution, variable 'a' represents a different quantity. On top of that, I make the work same so that I can represent it as 1 complete work. This means that I take an extra step before forming the equation. Other solutions in which 40 is in the numerator simplify the equation later.
Finally, you can use either method. Both are the same.

Compare my solution with Bunuel's solution to understand this:

Let the time needed to print 40 pages for printer A be a minutes, so for printer B it would be a−4 minutes.
For printer A, Work done = 40 pages, time taken = a mins
For printer B, work done = 40 pages, time taken = a - 4 mins
For both printers together, work done is 50 pages and time taken is 6 mins.

\(\frac{40}{a}+\frac{40}{a-4}=\frac{50}{6}\)

Compare this with:

We need time taken by printer A to print 80 pages. (Let's say this is 'a' mins)
We know A takes 4 more mins for 40 pages so it will take 8 more minutes for 80 pages.
Together, they print 50 pages in 6 mins so they will print 80 pages in 6*50/80 = 48/5 mins.
Now, I can say that printing 80 pages is 1 complete work.

\(\frac{1}{a} + \frac{1}{(a - 8)} = \frac{5}{48}\)

or in other words, if you ignore the 1 complete work concept:
\(\frac{80}{a} + \frac{80}{(a - 8)} = \frac{80}{48/5}\)
80 gets cancelled and you are left with the previous equation.
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Re: It takes printer A 4 more minutes than printer B to print 40   [#permalink] 10 Jan 2016, 21:29

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