After taxes, Sally takes home a salary of J = $5000 every month.

We can solve this by backtracking starting from the known numbers.

Step 1: Calculate L
33.33% of L Spent on Gifts , 40% of L spent on savings , balance = 26.67%
We know that 26.67% is 200$, So 100% of L is 750$

Step 2: Calculate K
K is twice the amount of L.
L=750 , so K = 1500.

Step 3: Calculate P
P % is the amount paid for Rent and Bills = 5000$-1500$ = 3500$
3500$ is 70% of 5000.

Ans: E

Instead of solving the problem algebraically, let’s make an educated guess for the value of P. Let’s guess P = 40 and check whether it satisfies all the criteria of the problem.

If P = 40, Sally pays 0.4 x 5000 = $2000 for rent and all her fixed bills, leaving her with 5000 - 2000 = $3000 (i.e., K = 3000). She then spends ½ x 3000 = 1500 on groceries, leaving her with 3000 - 1500 = $1500 (i.e., L = 1500). She then spends ⅓ x 1500 = $500 on gifts and puts ⅖ x 1500 = $600 into her savings account. This would leave her with 1500 - 500 - 600 = $400 for miscellaneous expenses. However, since we are given that she is left with only $200, we see that 40 is not the correct answer.

Let’s try P = 70.

If P = 70, Sally pays 0.7 x 5000 = $3500 for rent and all her fixed bills, leaving her with 5000 - 3500 = $1500 (i.e., K = 1500). She then spends ½ x 1500 = 750 on groceries, leaving her with 1500 - 750 = $750 (i.e., L = 750). She then spends ⅓ x 750 = $250 on gifts and puts ⅖ x 750 = $300 into her savings account. This would leave her with 750 - 250 - 300 = $200 for miscellaneous expenses. Since this matches the dollar amount of her miscellaneous expenses, we see that P = 70.

(Note: The reason we try 40 and 70 first is because of the dollar amount for K. Notice that K would be 3000 and 1500, respectively, which are numbers for which we can easily take ½, ⅓ and ⅖ of. Had we used P = 50, we see that K would have been 2500, which is not easily divided by 3.)

Answer: E

This question is conceptually simple. However it does take some additional time than an average question of this difficulty.

Interpret the given question as equations.

Statement 1: She pays P percent of this to her rent and all her fixed bills each month, leaving her with K left.
\(5000 - \frac{P}{100}5000 = K\)

Statement 2: She spends half of K on groceries, leaving her with L left.
She spends half of K and the remaining half is equal to L
\(\frac{K}{2} = L\)

Statement 3: If she spends \(\frac{1}{3}\) of L on gifts and puts \(\frac{2}{5}\)​​ of L into her savings account, this would leave her with $200 for miscellaneous expenses.

\(L - \frac{1}{3}L - \frac{2}{5}L = 200$\)

Taking LCM, we get \(\frac{4L}{15} = 200\)
\(L = 750\)

\(L = \frac{K}{2} --> K = 1500\)

\(5000 - \frac{P}{100}5000 = K\)
\(5000 - \frac{P}{100}5000 = 1500\)
\(5000 - 1500 = \frac{P}{100}5000\)
\(3500 = \frac{P}{100}5000\)
\(\frac{350000}{5000} = P\)

\(P = 70\)

Option(E) is the correct answer

Given: After taxes, Sally takes home a salary of J = $5000 every month. She pays P percent of this to her rent and all her fixed bills each month, leaving her with K left. She spends half of K on groceries, leaving her with L left. If she spends \(\frac{1}{3}\) of L on gifts and puts \(\frac{2}{5}\)​​ of L into her savings account, this would leave her with $200 for miscellaneous expenses.

Asked:What is the value of P?

Salary J = $5000
Rent = P%*J = 50P
K = 5000 - 50P = 50 (100-P)
Groceries = 50/2 (100-P) = 25(100-P)
L = 25(100-P)

Spend on gifts = L/3
Savings =2L/5

Miscellaneous expenses = L - L/3 - 2L/5 = (15-5-6)L/15 = 4L/15 = 4*25(100-P)/15 = $200
100 - P = 30
P = 70

IMO E

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