Question 244885
best way to solve this is to assign different letters to each occurrence.


you build your equations up and then you solve backwards.


here's how it works:


your problem as states is:


Some people got on a bus. At the rest stop, two-fifths of those people got off and three-fifths of that original number got on. At the second stop, one-half of the people got off, and one-third of the number left on the bus got on. At the last stop, three-quarters of the people got off the bus, leaving 5 people on the bus. How many people were on the bus before the bus reached its first stop?


a = number of people who got on the bus initially.
b = number of people who were left on the bus after people got off at the first stop.
c = number of people who were left on the bus after people got on at the first stop.
d = number of people who were left on the bus after people got off at the second stop.
e = number of people who were left on the bus after people got on at the second stop.
f = number of people who were left on the bus after people got off at the last stop.


your equations become:


b = a - (2/5 * a)


c = b + (3/5 * a)


d = c - (1/2 * c)


e = d + (1/3 * d)


f = e - (3/4 * e)


f = 5


you are left with f = 5


solve backwards.


f = e - (3/4 * e) becomes 5 = 1/4 * e becomes e = 20


e = d + (1/3 * d) becomes 20 = 4/3 * d becomes d = 15


d = c - (1/2 * c) becomes 15 = 1/2 * c becomes c = 30


c = b + (3/5 * a) becomes 30 = b + (3/5 * a) becomes b = 30 - (3/5 * a)


b = a - (2/5 * a) becomes 30 - (3/5 * a) = a - (2/5 * a)


add (3/5 * a) to both sides of this equation to get:


30 = a - (2/5 * a) + (3/5 * a) 


simplify to get:


30 = a + (1/5 * a) = 6/5 * a


multiply both sides of equation by 5 and divide both sides of equation by 6 to get:


30 * 5 / 6 = a


simplify to get:


a = 25


that's your answer.


you prove it's true by substituting in the original equation to confirm the answer is true (people left on the bus should be equal to 5).


start with 25.


at the first stop:


25 - 2/5 * 25 = 25 - 10 = 15 people left on the bus.
15 + 3/5 * 25 = 15 + 15 = 30 people left on the bus.


at the second stop:


30 - 1/2 * 30 = 30 - 15 = 15 people left on the bus.
15 + 1/3 * 15 = 15 + 5 = 20 people left on the bus.


at the third stop:


20 - 3/4 * 20 = 20 - 15 = 5 people left on the bus.


looks like we're good.