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Math Expert V
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Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Question Stats: 66% (03:09) correct 34% (03:14) wrong based on 258 sessions

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Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

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Posts: 61385
Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

We need to find the time taken by machine A to make 40 widgets. It will be best to take the time taken by machine A to make 40 widgets as the variable x. Then, when we get the value of x, we will not need to perform any other calculations on it and hence the scope of making an error will reduce. Also, value of x will be one of the options and hence plugging in to check will be easy.

Machine A takes x hrs to make 40 widgets.

Rate of work done by machine A = Work done/Time taken = $$\frac{40}{x}$$

Machine B take 2 hrs less than machine A to make 20 widgets hence it will take 4 hrs less than machine B to make 40 widgets. Think of it this way: Break down the 40 widgets job into two 20 widget jobs. For each job, machine B will take 2 hrs less than machine A so it will take 4 hrs less than machine A for both the jobs together.

Time taken by machine B to make 40 widgets = x – 4

Rate of work done by machine B = Work done/Time taken = $$\frac{40}{(x - 4)}$$.

We know the combined rate of the machines is 25/3

So here is the equation:

$$\frac{40}{x} + \frac{40}{(x - 4)} = \frac{25}{3}$$

The steps till here are not complicated. Getting the value of x poses a bit of a problem.

Notice here that that the right hand side is not an integer. This will make the question a little harder for us, right? Wrong! Everything has its pros and cons. The 3 of the denominator gives us ideas for the values of x (as do the options). To get a 3 in the denominator, we need a 3 in the denominator on the left hand side too.

x cannot be 3 but it can be 6. If x = 6, $$\frac{40}{(6 - 4)} = 20$$ i.e. the sum will certainly not be 20 or more since we have $$\frac{25}{3} = 8.33$$ on the right hand side.

The only other option that makes sense is x = 12 since it has 3 in it.

$$\frac{40}{12} + \frac{40}{(12 - 4)} = \frac{10}{3} + 5 = \frac{25}{3}$$

If we did not have the options, we might have tried x = 9 too before landing on x = 12. Nevertheless, these calculations are not time consuming at all since you can get rid of the incorrect numbers orally. Making a quadratic and solving it is certainly much more time consuming.

Another method could be to bring 3 to the left hand side to get the following equation:
$$\frac{120}{x} + \frac{120}{(x - 4)} = 25$$

This step doesn’t change anything but it helps if you face a mental block while working with fractions. Try to practice such questions using these techniques – they will save you a lot of time.
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

Machine B takes T hours, Machine A takes=T+2 hours
In 1 hours, they can make= (20/T+20/T+2)
Therefore, in 3 hours, they can make= 3* (20/T+20/T+2)
Solving for T, T=4, T+2=6
Therefore, if A can takes 6 hours to make 20 widgets, he will take 6*40/20=12 hours to make 40 widgets
##### General Discussion
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Posts: 101
Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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2
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

Solution:

Let x and y be the no. of widgets made in one hour by A and B and n be the no. of hours taken by B to complete 20 widgets.
Then, ny=20 and (n+2)x=20 ==> y=20/n and x=20/(n+2) -->(1)
Given, 3(x+y) = 25 ==> 3*20(1/n + 1/(n+2)) = 25.
Solving, we get n=4

So, x=20/6.

Time taken for machine A to make 40 widgets = 2(Time taken for machine A to make 20 widgets) = 2(6) =12

Option, E.
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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4
2
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

CONCEPT: Calculate one hour work of everyone to do further calculation

Let,
Time taken by Machine A to make 20 widgets = t hours
i.e. Time taken by Machine B to make 20 widgets = (t-2) hours

Calculating One hour work of each of them

Widgets made by Machine A in t Hour = 20
Widgets made by Machine A in 1 Hour = 20/t

Similarly,
Widgets made by Machine B in 1 Hour = 20/(t-2)

i.e. Widgets made by Machine A and B together in 1 Hour = (20/t) + [20/(t-2)] = 20(2t-2)/t(t-2)

But (Given),
Widgets made by Machine A and B together in 3 Hour = 25
i.e. Widgets made by Machine A and B together in 1 Hour = 25/3

i.e. 20(2t-2)/t(t-2) = 25/3
i.e. 4(2t-2)/t(t-2) = 5/3
i.e. 12(2t-2)=5t(t-2)
i.e. 24t-24 = 5t^2 - 10t
i.e. 5t^2 - 34t + 24 = 0
i.e. 5t^2 - 30t - 4t + 24 = 0
i.e. (t-6)(5t-4) = 0
i.e. t = 6 or 4/5
but 4/5 is not possible as (t-2) will be Negative in that case
therefore, t=6

i.e. A takes 6 hours to make 20 widgets
i.e. A will take 12 hours to make 40 widgets

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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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2
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

took 4.5 mins but got it finally

Rate of B = 20/t
Rate of A = 20/(t+2)

for 25 widgets, time taken by B = 5t/4 = T1
for 25 widgets, time taken by A = 5(t+2)/4 =T2

using time for both working together as T1*T2/(T1+T2) = 3 and substituting T1 and T2 from above equations, we get a quadratic equation.
solving this: t = 4 hrs.

so, time for A to produce 20 widgets = 6 hrs
for 40 widgets it takes double the time so 12 hrs.

E
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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It almost took me to solve through equations and reach out to Answer E

Is there any shortcut or quicker method in order to save precious time on D Day
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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3
1
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

i approached this one by plugging in numbers...started with C.
if 40 are made in 8 hours, then 20 are made in 4 hours. so time of A is 4, and time of B is 2.
rate together: 20/4 + 20/2 = 5+10 = 15. so in 1 hour, together make 15 widgets. in 3 hours = 45. way too much. we can eliminate right away C, B, and A - because B and A reduces the time - the total # of widgets made will be even higher.
now between D and E -> try only one ..if it doesn't work, then the other one is the answer.
i picked E:
12h to make 40 widgets, and 6 hours to make 20. this is the time of A. time of B=4 hours.
20/6 + 20/4 = 10/3 + 20/4
find LCM of 3 and 4 = 12. multiply first by 4, and second by 3:
40+60/12 = 100/12
divide by 4:
25/3
so this is the rate given.

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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

....Assume if machine rates are equal...to produce 25 widgets...each one of them takes 6 hours...since we know that machine A is slower it takes more than 6 hours...To produce 20 widgets the gap between 2 m/c s is 2hours...for 25 widgets the gap 2.5... we see that (1/t+1/t+2.5=1/3..)...and t+2.5 have to greater than 7 and less than 8(for integers )..satisfy this(actual number is 7.5)....
to produce 40 widgets...it takes 40/25*(7+) hrs.......only answer is 12...satisfy this...
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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3
2
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

Here is how you can do it without any variables:

Together they make 25 widgets in 3 hrs
So to make 40 widgets, together they take 3*(40/25) = 4.8 hrs
So each individually will certainly take more than 4.8 hrs to make 40 widgets.

Machine A takes 2 extra hrs to make 20 widgets so it will take 4 extra hrs to make 40 widgets. So machine A will certainly take more than 8.8 hrs to make 40 widgets. So we need to pick out of (D) and (E) only.
Try (D) - If A takes 10 hrs, B takes 6 hrs and together they 1/(1/10 + 1/6) = 60/16 (less than 4 hrs)

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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Top Contributor
1
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

Time actually worked / time it takes to do the complete job = 1
It takes 4.8 hrs together to complete 40 widgets.
A takes 4 hours more than B to complete 40 widgets
4.8/A + 4.8/(A-4) = 1
A=12
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Hi Guys,

One of the methods I used to determine the answer (after taking way too long) was to see the following:

• 40n+40/n(n+2) = 25/3
• 40(n+1) = 25x

So 40(n+1) will be a multiple of 25. We can see pretty quickly the first few options this gives us: n = 4, n=9
Plugging these back into the denominator n(n+2) ... 4(4+2) = 24: 200/24 = 25/3
So we don't have to try n=9. We know n=4.
From there we know that it will take twice as long to do 40 widgets than it does to do 20. so 2 x (n+2) = Answer
2 * (4+2) = 12

Manager  B
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Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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I did the same calculations with solving a quadratic equation and it took me a while
There is a kind of shortcut which can be better option taking into account a tight GMAT time
Machine B rate 20/t
Machine A rate 20/(t+2)

let's write an equation to produce 25 widgets in 3 hours
(20/t + 20(t+2))* 3 = 25
60/t+ 60/(t+2) = 25

from now we can play a plug and guess game, we know that B has better performance so what number can be t to solve the equation for 25.
My second try was 4 and 6. Took me about a minute or a bit more for backward checking
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Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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VeritasPrepKarishma wrote:
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

Here is how you can do it without any variables:

Together they make 25 widgets in 3 hrs
So to make 40 widgets, together they take 3*(40/25) = 4.8 hrs
So each individually will certainly take more than 4.8 hrs to make 40 widgets.

Machine A takes 2 extra hrs to make 20 widgets so it will take 4 extra hrs to make 40 widgets. So machine A will certainly take more than 8.8 hrs to make 40 widgets. So we need to pick out of (D) and (E) only.
Try (D) - If A takes 10 hrs, B takes 6 hrs and together they 1/(1/10 + 1/6) = 60/16 (less than 4 hrs)

Dear VeritasPrepKarishma,

If all the options were more than 8.8, how would we solve this problem using this method? Should we test all the options?
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Mehemmed wrote:
VeritasPrepKarishma wrote:
Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Kudos for a correct solution.

Here is how you can do it without any variables:

Together they make 25 widgets in 3 hrs
So to make 40 widgets, together they take 3*(40/25) = 4.8 hrs
So each individually will certainly take more than 4.8 hrs to make 40 widgets.

Machine A takes 2 extra hrs to make 20 widgets so it will take 4 extra hrs to make 40 widgets. So machine A will certainly take more than 8.8 hrs to make 40 widgets. So we need to pick out of (D) and (E) only.
Try (D) - If A takes 10 hrs, B takes 6 hrs and together they 1/(1/10 + 1/6) = 60/16 (less than 4 hrs)

Dear VeritasPrepKarishma,

If all the options were more than 8.8, how would we solve this problem using this method? Should we test all the options?

$$\frac{1}{A} + \frac{1}{A + 4} = \frac{1}{4.8}$$

$$\frac{1}{A} + \frac{1}{A + 4} = \frac{5}{24}$$

Now note that if A is 10, A + 4 = 14. LCM would be 70 which is not a multiple of 24 so let's try another value.
If A = 12, A + 4 = 16. LCM would be 48, a multiple of 24 which is quite possible. Check it out.
and so on...
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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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Here is how we can do without converting to decimal or long derivation:
Say Machine A can make 40 widget it A hours. B can make them 4 hours less say in A-4 hrs.
A & B together, we can arrive at the needed formula as : 1/A + 1/(A-1)
A& B 's combined rate is given for 25 widget => 3 hours.
To easily convert that to what we need, i.e 40 widgets , first consider as they make 100 widgets (25*4) in => 3*4 or 12 hrs
So, to make 40 widgets together, they need (12/5)*2 = 24/5 hours
i.e the rate is 5/24
So we know the value of 1/A + 1/(A-1) = 5/24
Now look at answer choices 6,8 and 12 can lead to a multiple of 24 as denominator.

Substituting values we can see L.HS = R.H.S for A=12 => 1/12 + 1/8 = 5/24

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Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor  [#permalink]

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A takes 2 hours more than B to finish 20 jobs. Mathematically: 20/A = 20/B + 2
A and B working together can finish 25 jobs in 3 hours. Mathematically: 25/A + 25/B = 3
Solving the equations for A, (A = 100).
Hence, the time taken for A to finish 40 jobs = 12 days. Re: Machine A takes 2 more hours than machine B to make 20 widgets. If wor   [#permalink] 23 Nov 2019, 03:33
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