+70 Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 4 times as much money as Bob. If, on the other hand, she gives n dollars to Bob, then she will have 3 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?
i got stuck on the 'she gives n dollars to bob, then she will have 3 times as much money as bob'... my english isn't very good.
Let's assume the amount of money Alice has is A dollars, and the amount of money Bob has is B dollars.
According to the given information, if Alice receives n dollars from Bob, she will have 4 times as much money as Bob. This can be expressed as:
A + n = 4(B - n) ---(1)
Similarly, if Alice gives n dollars to Bob, she will have 3 times as much money as Bob:
A - n = 3(B + n) ---(2)
We can solve these equations to find the values of A and B.
Expanding equation (1), we get:
A + n = 4B - 4n A + 5n = 4B
Expanding equation (2), we get:
A - n = 3B + 3n A - 3n = 3B
Now we have a system of equations:
A + 5n = 4B ---(3) A - 3n = 3B ---(4)
To eliminate A, we can subtract equation (4) from equation (3):
(A + 5n) - (A - 3n) = (4B) - (3B) 8n = B
Substituting this value back into equation (4):
A - 3n = 3(8n) A - 3n = 24n A = 27n
So, the ratio of the amount of money Alice has (A) to the amount Bob has (B) is:
A/B = (27n)/(8n) = 27/8
Therefore, the ratio of the amount of money Alice has to the amount Bob has is 27:8.
You set up your equations correctly, but somehow you didn't solve them correctly.
A + n = (4B - 4n)..............(1)
A - n = (3B + 3n)..............(2), solve for A, B
From (1) above: A==4B - 4n - n
A==4B - 5n {sub this into (2) above
4B - 5n - n ==3B + 3n
4B - 6n == 3B + 3n
4B - 3B ==3n + 6n
B ==9n {sub this into (2) above}
A - n ==(3 * 9n) + 3n
A - n ==27n + 3n
A ==27n + 3n + n
A == 31n
Therefore, the ratio of their money BEFORE the exchange==A / B ==31n / 9n, or 31: 9
Let's assume the amount of money Alice has is denoted by A and the amount Bob has is denoted by B.
According to the given information:
If Alice receives n dollars from Bob, she will have 4 times as much money as Bob: A + n = 4(B - n).
If Alice gives n dollars to Bob, she will have 3 times as much money as Bob: A - n = 3(B + n).
Simplifying the equations, we have:
A + n = 4B - 4n
A - n = 3B + 3n
We can rewrite equation 1 as: A + 4n = 4B.
Adding equation 2 and the rewritten equation 1, we get: (A - n) + (A + 4n) = 3B + 3n + 4B 2A + 3n = 7B + 3n 2A = 7B
Dividing both sides by B, we have: 2A/B = 7
Therefore, the ratio of the amount of money Alice has to the amount Bob has is A/B = 7/2.
Let's assume the amount of money Alice has is denoted by A and the amount Bob has is denoted by B.
According to the given information:
If Alice receives n dollars from Bob, she will have 4 times as much money as Bob: A + n = 4B.
If Alice gives n dollars to Bob, she will have 3 times as much money as Bob: A - n = 3B.
We can solve this system of equations to find the values of A and B.
First, let's solve equation 1 for A: A = 4B - n.
Substituting this value of A into equation 2: 4B - n - n = 3B. 4B - 2n = 3B. B = 2n.
Now, substituting the value of B back into equation 1: A = 4(2n) - n, A = 8n - n, A = 7n.
Therefore, we have found that A = 7n and B = 2n.
The ratio of the amount of money Alice has to the amount Bob has is A/B = (7n)/(2n) = 7/2.
So, the ratio is 7:2.
