Hard question that I don’t know if you are able to solve

Question

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?

Answer 1

Let J = amount of money John was given

Let N = amount of money Nat was given

Let x = #of days for John to spend all but $2500 and Nat to spend it all

Let y = #of days for John to spend all but $1000 and Nat to spend it all

J - 50x = 2500 (John spends $50 a day for x days and still has $2500 left)

N - 100x = 0 (Nat spends $100 for same number of days as John, but has spent it all)

J - 100y = 1000 (etc.)

N - 50y = 0

Solve the equations involving N for N:

N = 100x

N = 50y

Using those expressions substitute back into the equations involving J to get two equations involving J and N.

J - N/2 = 2500 (this is because when N = 100x, N/2 = 50x)

J - 2N = 1000 (this is because when N = 50y, 2N = 100y)

Multiply 1st equation by 2 to clear the fraction:

2J - N = 5000

J - 2N = 1000

Now you have a system of two equations two unknowns use either substitution or addition/elimination to solve for J and N. I would probably use the 2nd equation to get to: J = 2N + 1000 and replace J in the 1st equation to reach:

2(2N + 1000) - N = 5000

Continue from there to solve for N, then substitute resulting value into J = 2N + 1000 to solve for J.

Do a check to make sure the answers make sense. This would include also solving for x and y.