Alice and bob each have a certain amount of money. if alice receives $n$ dollars from bob, then she will have 3 times as much money as bob. if, on the other hand, she gives $n$ dollars to bob, then she will have 2 times as much money as bob. if neither gives the other any money, what is the ratio of the amount of money alice has to the amount bob has?

Let x =Alice’s original amount of money,

y = Bob’s original amount of money

And n = the dollars they are referring to

So, the equation are:

If Alice receives dollars from Bob, then she will have 3 times as much money as Bob,

x + n = 3(y - n)

x +n = 3y – 3n

x = 3y-3n-n

x = 3y – 4n

And if Alice give dollars to Bob, the she will have 2 times as much money as Bob,

x – n = 2 (y +n)

x – n = 2y + 2n

x = 2y + 2n + n

x = 2y + 3n

As, x = x

So, 3y – 4n = 2y + 3n

3y = 2y + 3n + 4n

3y – 2y = 3n + 4n

Y = 7n

Find x in terms of n also,

x = 2y + 3n, replace y with 7n

x = 2(7n) + 3n = 14n + 3n

x = 17n

If neither gives the other any money, the ratio of the amount of money Alice has to the amount Bob has = x/y

Replace x and y with their values, x = 17n and y = 7n

= 17n / 7n

The ratio is 17:7

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