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Bunuel
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Mark and Kate individually take 12 hours more and 27 hours more, respe 26 Sep 2018, 05:17
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C
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55% (hard)

Question Stats:

68% (02:34) correct 32% (02:50) wrong based on 263 sessions

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Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together. How many hours do Mark and Kate take to complete the project, working together?

(A) 12
(B) 16
(C) 18
(D) 24
(E) 39
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C
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Mark and Kate individually take 12 hours more and 27 hours more, respe 27 Sep 2018, 04:30
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Bunuel
Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together. How many hours do Mark and Kate take to complete the project, working together?

(A) 12
(B) 16
(C) 18
(D) 24
(E) 39
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Say they both take T hrs when working together.

When Mark works independently, he takes 12 hrs to do what Kate does in T hrs.
Ratio of speed of Mark:Kate = T:12

When Kate works independently, she takes 27 hrs to do what Mark does in T hrs.
Ratio of speed of Mark:Kate = 27:T

\(\frac{T}{12} = \frac{27}{T}\)

\(T = 18 hrs\)
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Mark and Kate individually take 12 hours more and 27 hours more, respe 26 Sep 2018, 21:02
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Bunuel
Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together. How many hours do Mark and Kate take to complete the project, working together?

(A) 12
(B) 16
(C) 18
(D) 24
(E) 39
Show more


The statement becomes \(\frac{1}{x+12}\)+\(\frac{1}{x+27}\)=\(\frac{1}{x}\)

Substitute the choices and see what fits in...


(A) 12.....1/24+1/27=1/12.....no because 1/12 is 2 times 1/24
(B) 16.....1/28+1/41=1/16.....no,because 41 cannot get cancelled being prime
(C) 18......1/30+1/45=1/15....75/30*45=5/2*45=1/2*9=1/18..yes
(D) 24......1/36+1/51=1/24...again no
(E) 39..same way no

C
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Mark and Kate individually take 12 hours more and 27 hours more, respe 26 Sep 2018, 21:45
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*(1/x+12)+(1/x+27)= 1/x
*(x+27+x+12)/[(x+12)(x+27)]=1/x
*(2x+39)/[(x+12)(x+27)]=1/x
2x^2+39x=x^2+39x+324
*X^2-324=0
*X=18

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Mark and Kate individually take 12 hours more and 27 hours more, respe 27 Sep 2018, 13:30
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chetan2u
Bunuel
Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together. How many hours do Mark and Kate take to complete the project, working together?

(A) 12
(B) 16
(C) 18
(D) 24
(E) 39


The statement becomes \(\frac{1}{x+12}\)+\(\frac{1}{x+27}\)=\(\frac{1}{x}\)

Substitute the choices and see what fits in...


(A) 12.....1/24+1/27=1/12.....no because 1/12 is 2 times 1/24
(B) 16.....1/28+1/41=1/16.....no,because 41 cannot get cancelled being prime
(C) 18......1/30+1/45=1/15....75/30*45=5/2*45=1/2*9=1/18..yes
(D) 24......1/36+1/51=1/24...again no
(E) 39..same way no

C
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chetan2u why are you adding values in denominator, as per the question if they work together with +12 and+27 hours it, then working together it means that their times should be x-12 and x-27 respectively
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Mark and Kate individually take 12 hours more and 27 hours more, respe 29 Sep 2018, 18:02
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Bunuel
Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together. How many hours do Mark and Kate take to complete the project, working together?

(A) 12
(B) 16
(C) 18
(D) 24
(E) 39
Show more

We can let x = the number of hours it would take Mark and Kate to finish the project if they were working together. Thus, Mark’s rate by himself is (x + 12) hours, and Kate’s rate by herself is (x + 27) hours.

Let’s create the equation for their rates. Mark’s rate is 1/(x + 12), Kate’s rate is 1/(x + 27), and their combined rate is 1/x. Thus, we have:

1/(x+12) + 1/(x+27) = 1/x

Multiplying by x(x+27)(x+12) we have:

x(x+27) + x(x+12) = (x+27)(x+12)

x^2 + 27x + x^2 + 12x = x^2 + 39x + 324

x^2 = 324

x = 18

Answer: C
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Mark and Kate individually take 12 hours more and 27 hours more, respe 08 Oct 2018, 03:24
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Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together. How many hours do Mark and Kate take to complete the project, working together?

x = √(a*b)

x = time taken for Mark and Kate to do job together, a = the additional time Mark takes to complete the job alone (i.e T+12), b = the additional time Kate takes to complete job alone (i.e T +27)

x = √324
x = 18

This is the fastest way for these.

Got this method from ganand.
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Mark and Kate individually take 12 hours more and 27 hours more, respe 17 Nov 2022, 04:01
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Given: Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together.

Asked: How many hours do Mark and Kate take to complete the project, working together?

Let Mark and Kate individually take m & k hours to complete the work

Mark and Kate individually take 12 hours more and 27 hours more, respectively, to complete a certain project than what they would have taken to complete the same project working together.

m = 12 + mk/(m+k)
k = 27 + mk/(m+k)
m - 12 = k - 27 = mk/(m+k)
k = m + 15
m - 12 = m(m+15)/(2m+15)
(m-12)/m = (m+15)/(2m+15)
m/12 = (2m+15)/m
mˆ2 = 12(2m+15)
mˆ2 - 24m - 180 = 0
(m-30)(m+6) = 0
m = 30
k = m + 15 = 45
mk/(m+k) = 30*45/(30+45) = 18

IMO C
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Mark and Kate individually take 12 hours more and 27 hours more, respe 15 Feb 2026, 07:30
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