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.
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 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.
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.