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Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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22 00:00

Difficulty:   75% (hard)

Question Stats: 62% (02:43) correct 38% (02:47) wrong based on 305 sessions

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Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

Originally posted by vscid on 11 Jan 2009, 17:31.
Last edited by Bunuel on 02 Jul 2015, 13:38, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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vscid wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

We can avoid the entire cumbersome calculation of fraction by choosing a RIGHT NUMBER for time and x

Let, x = 24 Units
Let, Time taken by Machine O to produce x(=24) units = 12 mins

i.e. Time taken by Machine O to produce x(=24) units = 12 mins
i.e. Time taken by Machine N to produce x(=24) units = (2/3)*12 = 8 mins
i.e. Time taken by Machine M to produce x(=24) units = (3/4)*8 = 6 mins

Machine O produces 24 units in 12 mins

i.e In 1 min time, the No. of Units produced by Machine O = 24/12 = 2 Units
i.e In 1 min time, the No. of Units produced by Machine N = 24/8 = 3 Units
i.e In 1 min time, the No. of Units produced by Machine M = 24/6 = 4 Units

i.e. Total Units Products by all three machine (M, N and O) together) = 2+3+4 = 9 units

Fraction of Total Work Done by Machine N (in 1 Min) = Units Produced by N / Total Units produced = 3/9 = 1/3

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Re: Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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vscid wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

For combined rates problems, if you can work out what each machine does in the same amount of time, the problem will almost always become easy. Since x is unimportant here, rather than talk about 'producing x units', I'll simply talk about 'completing one job'. Say it takes M t hours to complete one job. Then we know, from the ratios provided:

M completes one job in t hours
N completes one job in 4t/3 hours
O completes one job in 2t hours

We can easily work out what each would do in 4t hours:

M completes 4 jobs in 4t hours
N completes 3 jobs in 4t hours
O completes 2 jobs in 4t hours

So if they work together for the same amount of time, M does 4/9 of the work, N does 3/9 = 1/3 of the work, and O does 2/9 of the work. The answer is 1/3.
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Re: Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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Exactly ! For combined work problems we cannot simply compare ratios as done in the first explanation.
Thanks Ian.

Now I did the following way: (reverse method)
Let O take t hours
Let N take (2/3)*t hours
Let M take 2/3t(3/4) = (1/2)*t hours

Now using the formula: 1/M + 1/N + 1/O = 1/T ( T = combined hours )

we get, (9/2t) = 1/T.
t = 9T/2 => N = (2/3)*t = (2/3)*(9T/2)

Hence, N = 3 T .
Now, the question is to find fraction of total output T that is produced by N,
so do we have to find T/N ?? or N/T ? Also please let me know if the above approach is wrong.
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Re: Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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2
We will take all rates in terms of N machine.

Suppose N Machine takes ’n’ Hours
Therefore M machine takes 3/4n hours and O machine takes 3/2 n Hours (n=3/2O)

All machines together in 1 hour will do= 4x/3n+x/n+2x/3n= 3x/n

So all three machines works at the rate of 3x/n and machine N works at the rate of x/n.

Output produced by machine N/Total output= x/n/3x/n= 1/3
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Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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vscid wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

Another way to look at the problem..

Time ratio for M and N

M:N = 3:4....Efficiency(or work done in the same time) Ratio will be the inverse of this..m:n = 4:3

Similarly..Eff. Ratio for N and O
n:o = 3:2

Overall
m:n:o = 4:3:2

Fraction of work done by N?

$$\frac{3}{4+3+2}$$

$$=\frac{1}{3}$$

(B)

Originally posted by ShashankDave on 17 Jul 2016, 06:40.
Last edited by ShashankDave on 18 Jul 2016, 02:53, edited 1 time in total.
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Re: Machine M can produce x units in 3/4 of the time it takes machine N to  [#permalink]

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vscid wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

Let the time taken be as under -

O = 12
N = 8
M = 6

Assume total Work be 24 units...

Thus efficiency of the 3 machines are as under -

O = 2
N = 3
M = 4

Total efficiency is 9 ( ie, 2 + 3 + 4 ) units/hr

So, work done by N is 3/9 = 1/3

Hence, answer will be (B) 1/3

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Machine Mean produce x units in 3/4 of the time it takes machine N to  [#permalink]

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Bunuel wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N ?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

If Machine N takes 120 minutes to produce x units, machine M will take $$\frac{3}{4}*120 = 90$$ minutes.
Similarly, as Machine N will take $$\frac{2}{3}$$rd of the time to produce the units machine O produces.
$$120 = \frac{2}{3}*x$$ -> $$x = 120*\frac{3}{2} = 180$$ minutes

If 360 units are being produced,
machine N will produce 3 units in a minute,
machine M will produce 4 units in a minute,
and machine O will produce 2 units in a minute.

Therefore, machine N will produce $$\frac{3}{3+4+2} = \frac{3}{9} = \frac{1}{3}$$ of the total units(Option B)
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Re: Machine Mean produce x units in 3/4 of the time it takes machine N to  [#permalink]

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Bunuel wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N ?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33
Attachment: MACHINES M, N AND O.PNG [ 2.28 KiB | Viewed 2380 times ]

Let the total work be 24 units so, efficiency of the machines are
Attachment: MNO Efficiency.PNG [ 2.91 KiB | Viewed 2381 times ]

Since all the 3 machines are working simultaneously work done by N is $$\frac{3}{(2+3+4)}$$ = $$\frac{1}{3}$$ , Answer must be (B)
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Re: Machine Mean produce x units in 3/4 of the time it takes machine N to  [#permalink]

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Bunuel wrote:
Machine M can produce x units in 3/4 of the time it takes machine N to produce x units. Machine N can produce x units in 2/3 the time it takes machine O to produce x units. If all three machines are working simultaneously, what fraction of the total output is produced by machine N ?

(A) 1/2
(B) 1/3
(C) 4/13
(D) 8/29
(E) 6/33

Let t = the time it takes machine O to produce x units; then the time it takes machine N to produce x units = (2/3)t, and the time it takes machine M to produce x units = (3/4)(2/3)t = (1/2)t.

Therefore, the rate of M is x/[(1/2)t] = 2x/t, the rate of N is x/[(2/3)t] = 3x/2t, and the rate of O is x/t. Their combined rate is 2x/t + 3x/2t + x/t = 4x/2t + 3x/2t + 2x/2t = 9x/2t.

If all three machines are working simultaneously, the fraction of the total output that is produced by machine N is:

(3x/2t)/(9x/2t) = 3x/9x = 1/3

Alternate Solution:

Let’s denote the number of units produced by machine O in 12 hours as y.

Since machine N can produce the same number of units in 2/3 of the time, machine N can produce y units in 8 hours.

Since machine M can produce the same number of units in 3/4 of the time necessary for machine N, machine M can produce y units in 6 hours.

Now, let’s suppose all the machines ran for 24 hours. In 24 hours, machine M will produce 4y units; machine N will produce 3y units, and machine O will produce 2y units. The total number of units produced is 4y + 3y + 2y = 9y. The number of units produced by machine N is 3y, which is (3y)/(9y) = 1/3 of all the units produced.

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_________________ Re: Machine M can produce x units in 3/4 of the time it takes   [#permalink] 07 Aug 2019, 02:30

# Machine M can produce x units in 3/4 of the time it takes machine N to  