The ratio of Anson's savings to Billy’s savings became 3:11 after Anson gave 1/10 of his savings to Billy. Both later spent the same amount of money at a book fair. In the end, the ratio of Anson’s savings to Billy’s savings became 1:9 and Billy had $264 more than Anson. What was the ratio of Anson's savings to Billy’s savings at first? How much savings did Anson had at first?
To solve this problem, we can use a system of equations. Let's start by letting A be Anson's savings and B be Billy's savings before Anson gave 1/10 of his savings to Billy.
From the first sentence, we know that:
A/B = 3/11
After Anson gave 1/10 of his savings to Billy, Anson had 9/10 of his original savings left and Billy had 11/10 of his original savings:
Anson's new savings = 9/10A
Billy's new savings = 11/10B
We also know that they spent the same amount of money at a book fair, so:
Anson's new savings / (Anson's new savings + x) = 1/9
Billy's new savings / (Billy's new savings + x) = 8/9
where x is the amount they spent at the book fair. We can solve for x by setting Anson's new savings equal to Billy's new savings:
9/10A + x = 11/10B + x
9A - 11B = -20x/1
We also know that Billy had $264 more than Anson after the book fair, so:
Billy's new savings - Anson's new savings = $264
11/10B - 9/10A = $264
22B - 18A = $5280/1
Now we have two equations with two variables:
9A - 11B = -20x
22B - 18A = $5280
We can solve for A/B by eliminating one variable. Multiplying the first equation by 2 and adding it to the second equation gives:
18A - 22B + 44B - 36A = $5280 - 40x
-18A + 22B = -20x
22B - 18A = $5280
Simplifying:
4B = $5280 + 20x
-18A + 22B = -20x
Substituting the first equation into the second:
-18A + (264 + 9/11*$5280/4) = -20x
Solving for x, we get:
x = $990
Substituting x back into the equation for A/B:
4B = $7440
A/B = 5/16
Therefore, the ratio of Anson's savings to Billy's savings at first was 5:16.
To solve this problem, we can use a system of equations. Let A be Anson's initial savings and B be Billy's initial savings. After Anson gives 1/10 of his savings to Billy, their new savings ratios become 3:11. This means that:
A/(B + 1/10*A) = 3/11
Multiplying both sides by B + 1/10*A, we get:
A = (3/11)*(B + 1/10*A)
Simplifying this equation, we get:
110A = 33B + 3A
107A = 33B
B = (107/33)A
After they both spend the same amount of money at a book fair, their new savings ratio becomes 1:9 and Billy has $264 more than Anson. This means that:
A - x = (1/10)*(B - x) (since they both spent the same amount)
where x is the amount they spent.
Also,
A - x = (1/10)*(B - x) - 264 (since Billy has $264 more than Anson)
Substituting B = (107/33)A into these equations and simplifying, we get:
x = (11/13)A
Solving for A using either equation, we get:
A = $110
Therefore, Anson had $110 in savings at first.