Machine K and Machine N, working simultaneously and independently at

Question

Machine K and Machine N, working simultaneously and independently at their respective constant rates, process 2/3 of the shipment of a certain chemical product in 1.6 hours. Then machine K stopped working, and Machine N, working alone at it's constant rate, processed the rest of the shipment in 2 hours. How many hours would it have taken Machine K, working alone at its constant rate, to process the entire shipment?

A. 3.8
B. 4.0
C. 4.8
D. 5.4
E. 6.0

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Official Answer

B

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gmatophobia · Accepted Answer

Machine K and Machine N, working simultaneously and independently at their respective constant rates, process 2/3 of the shipment of a certain chemical product in 1.6 hours.

Inference : To process  \frac{2}{3}^{	ext{rd}}  of the shipment, Machine K and Machine N, working simultaneously and independently at their respective constant rates, take  1.6  hours. Hence, to process the entire shipment the machines would have taken  \frac{3}{2}^{	ext{rd}}  the time taken to process  \frac{2}{3}^{	ext{rd}}  of the shipment.

To process  \frac{2}{3}^{	ext{rd}}  unit of the shipment -----------------  1.6  hours

To process  1  unit of the shipment -----------------  1.6 * \frac{3}{2} = 2.4  hours

Then machine K stopped working, and Machine N, working alone at it's constant rate, processed the rest of the shipment in 2 hours

Inference : Hence, machine N processed the remaining  \frac{1}{3}^{	ext{rd}}  shipment in 2 hours. Therefore, to process the entire shipment alone, machine K would have taken thrice the time taken to process  \frac{1}{3}^{	ext{rd}}  shipment.

To process  \frac{1}{3}^{	ext{rd}}  unit of the shipment Machine K, working alone, took -----------------  2  hours

To process  1  unit of the shipment Machine K, working alone, will take -----------------  2 * 3 = 6  hours

If Machine N, working alone at its constant rate, takes  T_1  hours to complete one unit of task and Machine K, working alone at its constant rate, takes  T_2  hours to complete one unit of the same task, the time taken by Machine K and Machine N, working simultaneously and independently at their respective constant rates, to complete one unit of task is given by -

T_{	ext{together}} = \frac{T_1 * T_2}{T_1 + T_2}

In this question -

T_{	ext{together}} = 2.4  hours
 
  T_1 = 6  hours
 
  T_2 =  ?

Substituting the values in the above formula

2.4 = \frac{6T_2}{6 + T_2}

14.4 + 2.4T_2 = 6T_2

3.6T_2 = 14.4

T_2 = \frac{14.4}{3.6} = 4.0

Option B

Bunuel · Answer

Machine N, working alone, in 2 hours processed  1 - \frac{2}{3} = \frac{1}{3}  of the shipment. Hence, in 1.6 hours, it processed  (time)*(rate) = 1.6*\frac{(\frac{1}{3})}{2} = \frac{4}{5}*\frac{1}{6} = \frac{4}{15}  of the shipment.

Thus, in 1.6 hours, the remaining  \frac{2}{3} - \frac{4}{15} = \frac{2}{5}  of the shipment was processed by Machine K. Therefore, to process the entire shipment alone, Machine K would take  \frac{1}{(\frac{2}{5})}*1.6 = \frac{5}{2}*\frac{8}{5} = 4  hours.

Answer: B

sachi-in · Answer

2/3 work -- A + B takes -- 60 + 0.6*60 = 96 min
1/3 work -- B takes -- 120 min
1/3 work -- A + B takes -- 96/2 = 48 min

considering rate of work per minute for individual and combined rate (for completing 1/3 work) we have:
1/120 + 1/A = 1/48
1/A = 1/48 - 1/120
1/A = 3/240

Hence to finish 1/3 work A takes 240/3 = 80min
Total work is finished by A in 3*80/60 = 4hrs

VivekSri · Answer

OA is B

Manbhav7 · Answer

One Easier way - 
Let work done = 30 units >> 2/3 of work done = 20 units
Combined Rate - Work done/ Total Time taken = 20/ 1.6 h = 12.5 units/h -----(1)

We know, N completed remaining 1/3 i.e. (30-20 = 10 units of work) in 2 hrs >> Rate of N = 10/2 = 5 Units/h

Now Rate of K = Combined Rate - Rate of N = 12.5 - 5 = 7.5 (From 1)

Time of K = Work/ Rate = 30 Units/7.5 = 4 hours

egmat · Answer

Let's break this down step by step:

Step 1: Find Machine N's rate (start with the easiest information)

Notice that Machine N works alone for the last part—it completes  \frac{1}{3}  of the shipment in 2 hours. This gives us clean information about N's rate.

Machine N's rate =  \frac{1}{3} \div 2 = \frac{1}{6}  of the shipment per hour

Think of it this way: if N can do  \frac{1}{3}  of the job in 2 hours, then in 1 hour it does half of that amount.

Step 2: Find the combined rate when both machines work together

Both machines together complete  \frac{2}{3}  of the shipment in 1.6 hours.

Here's a useful trick: convert 1.6 to a fraction first:  1.6 = \frac{8}{5}

Combined rate =  \frac{2}{3} \div \frac{8}{5} = \frac{2}{3} 	imes \frac{5}{8} = \frac{10}{24} = \frac{5}{12}  of the shipment per hour

Step 3: Find Machine K's individual rate

Here's the key insight you need to see: when two machines work together, their combined rate equals the sum of their individual rates.

Combined rate = K's rate + N's rate

So: K's rate = Combined rate - N's rate

K's rate =  \frac{5}{12} - \frac{1}{6}

Convert to common denominator:  \frac{1}{6} = \frac{2}{12}

K's rate =  \frac{5}{12} - \frac{2}{12} = \frac{3}{12} = \frac{1}{4}  of the shipment per hour

Step 4: Calculate time for K to complete entire shipment

If K completes  \frac{1}{4}  of the job per hour, then to complete 1 entire job:

Time =  1 \div \frac{1}{4} = 4  hours

Answer: B (4.0 hours)

The approach I've shown you here gets you to the answer, but there's actually a more systematic framework for tackling all work rate problems efficiently. You can learn the complete framework that works for all work rate problems and discover the common mistakes you need to avoid (like reciprocal confusion and rate calculation errors) in the  detailed solution on Neuron by e-GMAT . You can also practice with  comprehensive explanations for other official GMAT questions here  for structured preparation with detailed analytics.

Hope this helps!

rikros · Answer

That's the easiest and fastest way to solve it imo:

rate = work/time and k and n are rates for Machine K and N respectively.

(k+n)*16/10=2/3 therefore (k+n)=2/3*10/16=5/12

n = 1/3*1/2 (1/3 of the shipment in 2hrs) = 1/6

and now if k+n=5/12 and n=1/6; k = 5/12-1/6=1/4

1/4 = 1(entire shipment)/hrs --> 4 hrs

nihilre · Answer

Rate of K = 1/6

(1/N + 1/K) 1.6 + (1/N)2 = 1

Subsititute and solve for N

Amity007 · Answer

Tried doing little quickly. Let me know if my method is wrong
Since, N worked alone for 1-2/3 of the work i.e. 1/3. It's rate would be 1/3 divide by 2. i.e. 1/6 
Now K and N completed 2/3 of the work in 1.6 h.
then total rate for K and N is 2/3 divide by 1.6. Rate for k is   - 1/6 Will come as 2*10/3*16 -1/6
so rate for K is 3/12 i.e. 1/4. so total hours for K is 4.

Machine K and Machine N, working simultaneously and independently at

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