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?

2 answers

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.