Emily receives her paycheck. She spends 1/6 on food. She spends 3/5 of remaining amount on house payment. She spends 1/3 of what is left on her other bills. She spends 1/4 of the remaining money for entertainment. This activity leaves her with $150. What was her original take home pay for the month?

To solve this problem, we need to work backwards and determine the original take-home pay by reversing the steps Emily took with her paycheck.

Let's denote Emily's original take-home pay as "P".

1. Emily spends 1/6 of her take-home pay on food. Therefore, the amount remaining after this expense is 5/6 of P.
remaining amount after food = (5/6)P

2. She spends 3/5 of the remaining amount on her house payment. After this expense, the amount remaining is 2/5 of the previous amount.
remaining amount after house payment = (2/5) * (5/6)P = (2/6)P = (1/3)P

3. Next, Emily spends 1/3 of what is left on her other bills. After this expense, the amount remaining is 2/3 of the previous amount.
remaining amount after other bills = (2/3) * (1/3)P = (2/9)P

4. Finally, Emily spends 1/4 of the remaining money on entertainment, leaving her with $150. So, the remaining amount after entertainment is 3/4 of the previous amount.
remaining amount after entertainment = (3/4) * (2/9)P = (6/36)P = (1/6)P

Now, we know that the remaining amount after all expenses is $150. Setting up an equation:

(1/6)P = $150

To solve for P, we can multiply both sides of the equation by 6:

P = ($150 * 6) = $900

Therefore, Emily's original take-home pay for the month was $900.

Let's break down the problem step by step:

1. Emily spends 1/6 of her paycheck on food.
- This means she has 1 - 1/6 = 5/6 of her paycheck remaining.

2. Emily spends 3/5 of the remaining amount on house payment.
- This means she has (5/6) - (3/5)(5/6) = (5/6) - (15/30) = (5/6) - (1/2) = 5/6 - 3/6 = 2/6 = 1/3 of her paycheck remaining.

3. Emily spends 1/3 of what is left on her other bills.
- This means she has (1/3)(1/3) = 1/9 of her paycheck remaining.

4. Emily spends 1/4 of the remaining money for entertainment.
- This means she has (1/9)(1/4) = 1/36 of her paycheck remaining.

5. Emily has $150 left.
- This means (1/36) of her paycheck is equal to $150.

To find her original take-home pay, we need to multiply $150 by the reciprocal of 1/36:

Original take-home pay = $150 / (1/36) = $150 * (36/1) = $5,400.

Therefore, Emily's original take-home pay for the month was $5,400.

paycheck: x

food: 1/6 x, leaving 5/6 x
house: 3/5 of 5/6 x = 1/2 x, leaving 1/3 x
bills: 1/3 of 1/3 x = 1/9 x, leaving 2/9 x
entertainment: 1/4 * 2/9 x = 1/18 x, leaving 1/6 x

so, 150 = 1/6 x
x = 900