Poisson wrote:On Wednesday, a certain number of passengers board an empty train leaving Washington, DC for Boston. In Baltimore, half of the passengers exit, and 20 additional passengers board the train. In Philadelphia, one-fourth of the passengers exit, and ten more passengers board the train. How many passengers did the train initially hold if there were 40 passengers on board after the train left Philadelphia?

A. 20
B. 25
C. 30
D. 35
E. 40

We can PLUG IN THE ANSWERS, which represent the initial number of passengers.
Since 1/2 the passengers exit in Baltimore, the correct answer choice must be EVEN.
Eliminate B and D.
When the correct answer choice is plugged in, there will be 40 passengers when the train departs Philadelphia.

C: 30
Since 1/2 exit in Baltimore, the number exiting in Baltimore = (1/2)(30) = 15.
Remaining passengers = 30-15 = 15.
After 20 passengers board in Baltimore, the new total = 15+20 = 35.
Since 1/4 exit in Philadelphia, the number exiting in Philadelphia = (1/4)(35) = 35/4.
Since the number of exiting passengers must be an INTEGER, eliminate C.

A: 20
Since 1/2 exit in Baltimore, the number exiting in Baltimore = (1/2)(20) = 10.
Remaining passengers = 20-10 = 10.
After 20 passengers board in Baltimore, the new total = 10+20 = 30.
Since 1/4 exit in Philadelphia, the number exiting in Philadelphia = (1/4)(30) = 15/2.
Since the number of exiting passengers must be an INTEGER, eliminate A.

The correct answer is E.

E: 40
Since 1/2 exit in Baltimore, the number exiting in Baltimore = (1/2)(40) = 20.
Remaining passengers = 40-20 = 20.
After 20 passengers board in Baltimore, the new total = 20+20 = 40.
Since 1/4 exit in Philadelphia, the number exiting in Philadelphia = (1/4)(40) = 10.
Remaining passengers = 40-10 = 30.
After 10 passengers board in Philadelphia, the new total = 30+10 = 40.
Success!

Poisson wrote:On Wednesday, a certain number of passengers board an empty train leaving Washington, DC for Boston. In Baltimore, half of the passengers exit, and 20 additional passengers board the train. In Philadelphia, one-fourth of the passengers exit, and ten more passengers board the train. How many passengers did the train initially hold if there were 40 passengers on board after the train left Philadelphia?

A. 20
B. 40
C. 60
D. 80
E. 100

The answer is B but I need help working through the logic of this problem. Thanks so much. Any suggestions for recognizing indirect equations?

Can be solved algebraically as follows:

0 + x - x/2 + 20 - 1/4(x/2 + 20) + 10 = 40
x/2 + 20 - x/8 - 5 + 10 = 40
3x/8 + 25 = 40
3x/8 = 15
x = (8*15)/3
x = 8*5
x = 40

800_or_bust wrote:

Poisson wrote:On Wednesday, a certain number of passengers board an empty train leaving Washington, DC for Boston. In Baltimore, half of the passengers exit, and 20 additional passengers board the train. In Philadelphia, one-fourth of the passengers exit, and ten more passengers board the train. How many passengers did the train initially hold if there were 40 passengers on board after the train left Philadelphia?

A. 20
B. 40
C. 60
D. 80
E. 100

The answer is B but I need help working through the logic of this problem. Thanks so much. Any suggestions for recognizing indirect equations?

Can be solved algebraically as follows:

0 + x - x/2 + 20 - 1/4(x/2 + 20) + 10 = 40
x/2 + 20 - x/8 - 5 + 10 = 40
3x/8 + 25 = 40
3x/8 = 15
x = (8*15)/3
x = 8*5
x = 40

Start with 0 since we are told the train was empty initially. Then set the number of passengers who entered in Washington, DC equal to x. So upon leaving Washington, DC, there are x passengers. Upon arrival in Boston, half the passengers leave, so in other words subtract x/2. Prior to leaving Boston, 20 passengers enter, so add 20. At this point, the train contains x/2 + 20 passengers. Upon arrival in Philadelphia, 1/4 of the passengers exit. In other words, 1/4(x/2 + 20) = x/8 + 5 passengers exit, so subtract this quantity. Finally, 10 new passengers enter the train at Philly so add this number. Then solve for x.

Say we started with p passengers.

After Baltimore, we've got .5p + 20.

After Philly, we've got .75*(.5p + 20) + 10. we know that's equal to 40, so

(3/4)*((1/2)p + 20) = 30, or p = 40.