Question

Henry and Sean have the same amount of income. If Henry spends $43 a day and Sean spends $14 a day, Sean will have $319 when Henry uses up all his income. If Henry spends $13 a day and Sean spends $43 a day, how much income will Henry have left when Sean uses up all his income?

Answer 1

if they each start with x, we have
x-14d = x-43d + 319
d = 11
so they each started with $473
473 - 11*13 = $330

Answer 2

Let x be the amount of income each have.
Let y be the number of days when Henry used up all his income
Then, we have,
x - 14y = 319 - (1)
and
x - 43y = 0 - (2)
Solving these two
y = 11
=> x = 473
income = $473
number of days = 11
Now when Henry spends $13 & Sean spends $43. Let z be the number of days when Sean uses up all his income.
=> 473 - 43z = 0
z = 11
Thus 473 - 13 * 11 = 330
Hence, Henry will have $330 when Sean uses up all his income