M13-28

Question

When working together, John and Jack can type 20 pages in one hour. If Jack were to increase his typing speed by 25%, they would be able to type 22 pages in one hour. What is the ratio of Jack's original typing speed to John's typing speed?

A. \(\frac{1}{3}\)
B. \(\frac{2}{5}\)
C. \(\frac{1}{2}\)
D. \(\frac{2}{3}\)
E. \(\frac{3}{5}\)

A.  \frac{1}{3}  
B.  \frac{2}{5}  
C.  \frac{1}{2}  
D.  \frac{2}{3}  
E.  \frac{3}{5}

Bunuel · Accepted Answer

Official Solution:

When working together, John and Jack can type 20 pages in one hour. If Jack were to increase his typing speed by 25%, they would be able to type 22 pages in one hour. What is the ratio of Jack's original typing speed to John's typing speed?

A. \(\frac{1}{3}\)
B. \(\frac{2}{5}\)
C. \(\frac{1}{2}\)
D. \(\frac{2}{3}\)
E. \(\frac{3}{5}\)

A.  \frac{1}{3}  
B.  \frac{2}{5}  
C.  \frac{1}{2}  
D.  \frac{2}{3}  
E.  \frac{3}{5}

Let the rate of John be  x  pages per hour and the rate of Jack be  y  pages per hour. The question asks to find the value of  \frac{y}{x} ?
 
As we can sum the rates and  rate*time=job \ done , we have  (x+y)*1=20 . So, we have  x+y=20 ; "If Jack were to increase his typing speed by 25%, they would be able to type 22 pages in one hour" can be expressed as  (x+1.25y)*1=22 . Therefore, we also have  x+1.25y=22 ;
 
Subtracting (1) from (2) gives  x+1.25y-(x+y)=22-20 , which results in  y=8 . Thus,  x=12 . Therefore,  \frac{y}{x}= \frac{8}{12}=\frac{2}{3} .
 
Alternatively, if by increasing Jack's rate by 25%, 2 more pages can be typed in one hour, we can directly write:  0.25y=2 , which gives  y=8  and  x=12 . Therefore,  \frac{y}{x}=\frac{8}{12}=\frac{2}{3} .

Answer: D

profileusername · Answer

How did we sum up the rates? From what I understand, we can invert the two rates and then add them up to make combined rate. Im sure Im missing some basic concept here. Please help me out.

Bunuel · Answer

THEORY There are several important things you should know to solve work problems: 1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems. time*speed=distance <--> time*rate=job \ done . For example when we are told that a man can do a certain job in 3 hours we can write: 3*rate=1 --> rate=\frac{1}{3} job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then 5*(2*rate)=1 --> so rate of 1 printer is rate=\frac{1}{10} job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then 3*(2*rate)=12 --> so rate of 1 printer is rate=2 pages per hour; So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate). 2. We can sum the rates. If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is rate_a=\frac{job}{time}=\frac{1}{2} job/hour and B's rate is rate_b=\frac{job}{time}=\frac{1}{3} job/hour. Combined rate of A and B working simultaneously would be rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6} job/hour, which means that they will complete \frac{5}{6} job in one hour working together. 3. For multiple entities: \frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T} , where T is time needed for these entities to complete a given job working simultaneously. For example if: Time needed for A to complete the job is A hours; Time needed for B to complete the job is B hours; Time needed for C to complete the job is C hours; ... Time needed for N to complete the job is N hours; Then: \frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T} , where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously. For two and three entities (workers, pumps, ...): General formula for calculating the time needed for two workers A and B working simultaneously to complete one job: Given that t_1 and t_2 are the respective individual times needed for A and B workers (pumps, ...) to complete the job, then time needed for A and B working simultaneously to complete the job equals to T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2} hours, which is reciprocal of the sum of their respective rates ( \frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T} ). General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job: T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3} hours. Theory on work/rate problems: https://gmatclub.com/forum/work-word-pro ... 87357.html All DS work/rate problems to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=46 All PS work/rate problems to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=66 Hope this helps

nisthagupta28 · Answer

I understand the solution given. 
Just have one query.

R * 1 hr = 20 page => the rate is 20 page/1hour or 20 p/hr
I took individual rates as 1/x + 1/y. Why is this incorrect?

Since time is 1 - the rates should be reciprocal of it. hence 1/x and 1/y. 
Please tell me what is it that I'm missing.

Bunuel · Answer

x and y in the solution provided are respective RATES of pages per hour of John and Jack, while in your solutions, x and y are respective TIMES in hour in which John and Jack could type 1 page. This is not incorrect but since we need the ratio of rates it's better to denote the rates by x and y from the beginning. Still if you proceed with your approach you'd get the correct answer:

1/x + 1/y = 20 and 1/x + 1/y*1.25 = 22 gives x = 1/12 hours and y = 1/8 hours. Thus rates would be 12 pages per hour and 8 pages per hour. Ratio = 8/12 = 2/3.

Hope it's clear.

AntonioGalindo · Answer

Bunuel , could you please expand the algebra? I can´t figure out how to solve the two equations

1/x + 1/y = 20 and 1/x + 1/y*1.25 = 22

Thank you.

Bunuel · Answer

(i)  \frac{1}{x} + \frac{1}{y} = 20

(ii)  \frac{1}{x} + \frac{1}{y}*1.25 = 22

Subtract (i) from (ii):

\frac{1}{y}*0.25=2 ;

y = \frac{1}{8} .

Plug back  y = \frac{1}{8}  in (i) to get  y = \frac{1}{12} .

Hope it helps.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

nivivacious · Answer

So if 25% = 2 ie (22-20)
then 100% = 8

20-8 = 12
Therefore 8:12 
ie 2:3

Robiacca · Answer

25% of Jack's original speed is equivalent to (22-20) 2 pages. That means originally Jack was typing 8 pages ((100/25)*2) among the 20 pages.

a191291r · Answer

Originally, both can type  20 pages/hr .

Given that Jack has increased his speed by  25 % (and John is still writing at same speed)

25 % of Jack's speed increase resulted in increase of  22-2=2 pages/hr

Therefore we can say that 100% of Jack's speed must be  \frac{100}{25}*2 = 8 pages/hr

So Original speed of Jack is  8 Pages/hr  and of John is  20-8=12pages/hr

Ratio = \frac{8}{12} = \frac{2}{3}

Bunuel  I solved it like this in my head as the calculations felt easy and answer was also correct. But is this the correct approach or was it a fluke?

Dbrunik · Answer

The solution above are very detailed and using great algebra, but this question can be solved much faster and with less additional variables. why add x and y and the write extra things when you dont need to ? heres how you should approach:

note that the question is asking for ((original jack)/ (john)).

it was 20, now its 22. jack is the only one that changed. What number, when increased by 25%, will be 2 greater? the answer is 8. this means the original rates are 8 and 12. so now, they are 10 and 12. that is 22.

no you know you simply just need to do 8/12. 2/3. easy.

adarsh2209 · Answer

I like the solution - it’s helpful.

rashis700 · Answer

I don’t quite agree with the solution. We have a formula for working together as xy/(x+y). If we solve this equation, we are getting half (1/2) as ratio.. please help me understand this disparity.

Bunuel · Answer

Not sure how you are applying that formula and how you are getting 1/2.

However, in the official solution, x and y represent John’s and Jack’s  rates  (pages per hour). Since rates add directly when working together, we use x + y = 20.

In the formula xy/(x + y), x and y represent  times  (hours taken individually to finish one whole job).

M13-28

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