+17 Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 7 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?
+904 Let's represent the amount of money Alice has as A and the amount Bob has as B. We can set up two equations based on the given information:
1. Alice receives n dollars from Bob: In this scenario, Alice's money increases by n, while Bob's money decreases by n. So, we have:
A + n = 7(B - n)
2. Alice gives n dollars to Bob: Here, Alice's money decreases by n, and Bob's money increases by n. So, we have:
A - n = 3(B + n)
Now, we want to find the ratio A/B without introducing the unknown value n. To achieve this, we can eliminate n by strategically manipulating the equations.
One approach is to add the two equations together:
(A + n) + (A - n) = 7(B - n) + 3(B + n)
Simplifying both sides:
2A = 10B Dividing both sides by B:
2A/B = 10
A/B = 10/2 = 5
Therefore, the ratio of Alice's money (A) to Bob's money (B) is 5:1
+1868 Let's create a system of equations to solve this problem .
Let's let A be the number of money Alice has and B be the amount of money Bob has. We have
If Alice receives n dollars from Bob, then she will have 3 times as much money as Bob.
\(A + n = 3 * (B - n) \)
If she gives n dollars to Bob, then she will have 2 times as much money as Bob.
\(A - n = 2 * (B + n) \)
Thus, we have the system
\(A + n = 3B - 3n → A = 3B - 4n → 3A = 9B - 12n \\ A - n = 2B + 2n → A = 3B + 3n → 4A = 12B + 12n\)
adding these two equations up, we get
\(7A = 17B → 7A / B = 17 → A / B = 17 / 7 \)
So, the answer is \(17/7\)
Thanks! :)
+129845 Let the original amount that Alice had = A Let the original amount that Bob had = B
1st scenario : Alice gets n dollars from Bob....her new amt = A + n
Bob loses n dollars.....his new amt times 7 = Alice's new amt
A + n = 7(B - n) → A + n = 7B - 7n → A + 8n = 7B (1)
2nd senario : Bob gets n dollars from Alice....her new amt = A - n
Bob gains n dollars and alice has 3 times as much as he does
A - n = 3(B + n) → A - n = 3B + 3n → A - 4n = 3B → 2A - 8n = 6B (2)
Add (1) , (2)
3A = 13B
A / B = 13 / 3
Proof : 3A = 13B → A = (13/3)B....sub this into (1)
(13/3)B + 8n = 7B
8n =(8/3)B multiply through by 3/8
3n = B
n= B/3
Using (2)
2A - 8(B/3) = 6B
2A = (26/3)B
A = (26/6)B
A = (13/3)B
A / B = 13 / 3
