oops this posted twice ignore this one

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 were John and Nat given each?

Let j = Johns money at start

Let n = Nats money at start

Let x = days until Nat spent all her money first run.

Let y = days until Nat spent all her money second run.

1. Run

\(\begin{array}{|rcll|} \hline j - x\cdot 50 &=& 2500 \\ j &=& 2500+50x\\\\ n - x\cdot 100 &=& 0 \\ n &=& 100\cdot x \\ \hline \end{array}\)

2. Run

\(\begin{array}{|rcll|} \hline j - y\cdot 100 &=& 1000 \\ j &=& 1000+100y \\\\ n - y\cdot 50 &=& 0 \\ n &=& 50y \\ \hline \end{array}\)

We set equal:

\(\begin{array}{|rcll|} \hline n = 50y &=& 100x \\ y &=& 2x \\\\ 1000+100y &=& 2500+50x \quad & | \quad -1000\\ 100y &=& 1500+50x \\ 100\cdot(2x) &=& 1500+50x \\ 200x &=& 1500+50x \quad & | \quad -50x\\ 150x &=& 1500 \quad & | \quad :150\\ x &=& \frac{1500}{150} \\ x &=& 10 \\\\ n &=& 100x \\ n &=& 100\cdot 10 \\ \mathbf{ n } & \mathbf{=} & \mathbf{$ 1000 }\\\\ j &=& 2500+50x \quad & | \quad x = 10\\ j &=& 2500+50\cdot 10 \\ j &=& 2500+500 \\ j &=& 3000 \\ \mathbf{ j } & \mathbf{=} & \mathbf{$ 3000 }\\\\ \hline \end{array}\)

John were given $3000 and Nat were given $1000

Let the amount of money John started with=M

Let the initial number of days =d

M - 50d =2500, M - 200d =1,000, solve for M, d

M=$3,000 original amount of money that John started with.

d=10 initial number of days.

Since Nan spent $100 per day initially, therefore she must have had:

10 x $100 =$1,000 to start with.

Since John spent $3,000 - $1,000 =$2,000 @ $100 per day, in the 2nd scenario, therefore it took him:

$2,000/$100=20 days to spend that money.