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Can you explain step by step how to approach this problem?

Together John and Jane eat 2/3 a pizza in 4 seconds.

How long would it take each of them to eat the whole pizza alone if Jane can eat a pizza 5 seconds faster than John?

2 Answers

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  • Alan
    Lv 7
    1 month ago

    Alternate Method 

    (4) Jane' Rate  +   4* (John's Rate) = 2/3 of a pizza 

    then,  multiply both sides by 3/2   

    6*Jane's Rate  + 6* John's Rate   = 1 pizza  

    so it takes them 6 seconds together 

    Jane's Rate = 1/Jane'S time to eat one pizza 

    John's rate = 1/ John's Time to eat one pizza 

    John's rate = 1/ (Jane's time + 5)   

    This gives 

    6/ (1/Jane_time)  + 6(1/ (jane's_time+5)  = 1

    Common denominator  

    6(jane's Time +5)  + 6(Jane's Time) / (jane's Time^2 + 5*Jane's Time) = 1

    jane'sTime^2  + 5*Jane's Time =  12*jane's Time +   30 

    jane's Time^2 -7*Jane's Time  - 30  = 0   

    (Jane's Time -10) (jane's Time +3) =0  

    Jane's time  =10 or -3 seconds 

    Jane's time = 10 seconds

    John's Time  = 15 seconds

    Checking 

    (4)(1/10) + 4(1/15)   = 4/10  + 4/15 =    12/30 +  8/30 = 20/30 = 2/3 

    6(1/10)  + 6(1/15)  =  6/10 +  6/15 =  18/30  + (12/30) = 1     

    Since my answer check out, the first answer has a mistake 

    It is right to here : 

    2(t² - 5t) = 24t - 60

    2t² - 10t = 24t - 60  (still right) 

    2t² - 24t + 60 = 0 (wrong) 

    should be 

    2t^2 -34t + 60 = 0   (divide by 2) 

    t² - 17t + 30 = 0  

    (t - 15)(t-2) = 0 

    so t = 15  or t = 2 for John  

    but t =2 is impossible 

    so John's time = 15 

    and 

    jane's time = 10  

  • 1 month ago

    work = rate * time

    Work here is the fraction of a pizza eaten in the given time.

    When they work together, they add their rates together.

    Let r = John's rate of pizza per time

    Let R = Jane's rate of pizza per time

    separately, it takes Jane 5 fewer seconds to eat a whole pizza than it takes for John to eat.  We need a new variable:

    Let t = time it takes for John to eat a pizza

    Then the time it takes for Jane to eat one is (t - 5).

    Now we can create two equations for r and R in terms of t:

    w = rt

    work here is 1 as it's the entire pizza.

    1 = rt and 1 = R(t - 5)

    1 / t = r and 1 / (t - 5) = R

    When they eat pizza together, we add the rates, so:

    1 / t + 1 / (t - 5)

    (t - 5) / [t(t - 5)] + t / [t(t - 5)]

    (t - 5 + t) / [t(t - 5)]

    (2t - 5) / [t(t - 5)]

    That's the rate when they work together.  We know they can eat 2/3 of a pizza in 4 seconds together, so:

    w = rt

    2/3 = (2t - 5) / [t(t - 5)] * 4

    We now have an equation with one unknown.  We can solve for t:

    2 = 12(2t - 5) / [t(t - 5)]

    2 = (24t - 60) / [t(t - 5)]

    2[t(t - 5)] = 24t - 60

    2(t² - 5t) = 24t - 60

    2t² - 10t = 24t - 60

    2t² - 24t + 60 = 0

    t² - 12t + 30 = 0

    I'll solve this by completing the square.  Subtracting both sides by 30, then adding 36 to both sides to complete the square:

    t² - 12t = -30

    t² - 12t + 36 = -30 + 36

    (t - 6)² = 6

    t - 6 = ± √6

    t = 6 ± √6

    there are two possible times.  We can now solve these for r and R so we can see which ones make sense (we can't have a negative rate, etc.)

    r = 1 / t and R = 1 / (t - 5)

    r = 1 / (6 - √6) and R = 1 / (6 - √6 - 5) and r = 1 / (6 + √6) and R = 1 / (6 + √6 - 5)

    r = (6 + √6) / [(6 - √6)(6 + √6)] and R = 1 / (1 - √6) and r = (6 - √6) / [(6 - √6)(6 + √6)] and R = 1 / (1 + √6)

    The first R here will end up being negative so we can throw out that solution, leaving the last answer to check:

    r = (6 - √6) / 30 and R = (1 - √6) / [(1 - √6)(1 + √6)]

    r = (6 - √6) / 30 and R = (1 - √6) / (1 - 6)

    r = (6 - √6) / 30 and R = (1 - √6) / (-5)

    r = (6 - √6) / 30 and R = (√6 - 1) / 5

    These are the exact rates.  Decimal approximations work out to be :

    r = 0.11835 and 0.28990 pizza per second

    If we use these rates to find the times, one should be 5 seconds faster than the other:

    w = rt

    1 = 0.11835t and 1 = 0.28990t

    1 / 0.11835 = t and 1 / 0.28990 = t

    8.44951 = t and 3.44947 = t

    It's not exact due to rounding, but the times are about 5 seconds off.

    So again the exact rates in terms of pizza per second for each person is:

    r = (6 - √6) / 30 and R = (√6 - 1) / 5

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