Working at his normal constant rate, John fills a certain type of

Question

Working at his normal constant rate, John fills a certain type of order in 3t whereas Mary, working at her normal constant rate, fills the same type of order in t. If at the same time each of them begins to fill an order at his or her normal constant rate, after how much time has passed in terms of t will Mary have remaining 1/3 the work to complete her order that John has remaining to complete his?

(A) \(\frac{t}{3}\)

(B) \(\frac{t}{2}\)

(C) \(\frac{2t}{3}\)

(D) \(\frac{3t}{4}\)

(E) \(t\)

Source: Marty Murray Coaching

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Official Answer

D

(A)  \frac{t}{3}

(B)  \frac{t}{2}

(C)  \frac{2t}{3}

(D)  \frac{3t}{4}

(E)  t

Source:  Marty Murray Coaching

MartyMurray · Answer

Working at his normal constant rate, John fills a certain type of order in 3t whereas Mary, working at her normal constant rate, fills the same type of order in t. If at the same time each of them begins to fill an order at his or her normal constant rate, after how much time has passed in terms of t will Mary have remaining 1/3 the work to complete her order that John has remaining to complete his?

(A) \(\frac{t}{3}\)

(B) \(\frac{t}{2}\)

(C) \(\frac{2t}{3}\)

(D) \(\frac{3t}{4}\)

(E) \(t\)

Source: Marty Murray Coaching

Show Answer

Official Answer

D

John's rate:  \frac{1}{3t}

Mary's rate:  \frac{1}{t}

Amount of time passed:  x

Amount of work John completes in  x  time:  \frac{x}{3t}

Amount of work John has remaining after  x  time has passed:  1 - \frac{x}{3t}

Amount of work Mary completes in  x  time:  \frac{x}{t}

Amount of work Mary has remaining after  x  time has passed:  1 - \frac{x}{t}

after how much time has passed in terms of  t  will Mary have remaining 1/3 the work ... that John has remaining

\frac{1}{3}(1 - \frac{x}{3t}) = 1 - \frac{x}{t}

1 - \frac{x}{3t} = 3 - \frac{3x}{t}

3t - x = 9t - 9x

8x = 6t

x = \frac{3t}{4}

(A)  \frac{t}{3}

(B)  \frac{t}{2}

(C)  \frac{2t}{3}

(D)  \frac{3t}{4}

(E)  t

Correct answer:   D

PocusFocus · Answer

As usual, nice problem Marty, it took me a while but did it correctly, here my solution:
Let's assume that work the amount of work is 3t

Time 
 Work 
 Rate 
 
  Jhon 
 3t 
 3t 
 1 
 
  Mary 
 t 
 3t 
 3

And (t') will be our incognita, after t' Jhon will advanced t' of work and Mary 3t' of work , now
Let's say that 3a is the amount of work remaining for John, so:

t' + 3a = 3t' + a --> a = t' --> replacing a = t' --> total work = 4t' = 3t,   t' = 3t/4

rak08 · Answer

MartyMurray , the framing of the question is weird.
it should clearly say " 1/3 of the remaining work of remaining work of John" which was not clear in the question.

MartyMurray · Answer

The wording is a little unusual, but I think there's only one possible meaning. Don't you?

Working at his normal constant rate, John fills a certain type of

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Working at his normal constant rate, John fills a certain type of

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