```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:

Some people got on a bus. At the rest stop, two-fifths of those people got off 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 off 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.

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

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).

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.

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