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

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  

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