Question 1163541
.
john and nat were given some money. if john spends $50 and nat spends $100 each day, john would still have $2500 left
while nat would have spent all her money. if john spends $100 and nat spends $50 each day, john would still have $1000 left 
while nat would have spent all her money. How much money john and nat were given each? 
~~~~~~~~~~~~~



<pre>
Let J be the amount John had initially and let x be the number of days in the 1-st scenario.

Let N be the amount Nat  had initially and let y be the number of days in the 2-nd scenario.


For the first scenario, we have these two equations

    J = 50x + 2500,     (1)

    N = 100x.           (2)


For the second scenario, we have these two equations

    J = 100y + 1000,     (3)

    N = 50y.             (4)


In all, we have 4 equations in 4 unknown  J, N, x and y, so we have a chance to solve it.



From eq(2), express x = {{{N/100}}}  and substitute it into eq(1).  You will have

    J = 50*(N/100) + 2500,   or  J = (1/2)*N + 2500,   or   2J - N = 5000  (5)


From eq(4), express y = {{{N/50}}}  and substitute it into eq(3).  You will have

    J = 100*(N/50) + 1000,    or  J = 2N + 1000,       or    J -2N = 1000  (6)



Thus, we reduced the problem to two equations in 2 unknowns

    2J -  N = 5000    (5)

     J - 2N = 1000    (6)


From eq(6), express  J = 2N + 1000  and substitute it into eq(5).  You will get

    2*(2N + 1000) - N = 5000,


which implies

    4N + 2000 - N = 5000

    3N            = 5000 - 2000 = 3000

     N            = 3000/3 = 1000.


Then from eq(5)

    2J = 5000 + 1000 = 6000

     J = 6000/2 = 3000.


<U>ANSWER</U>.  John had 3000;  Nat had 1000.
</pre>

Solved.