The ratio of the number of marbles Ryan had to the number of marbles Audrey had at first was 2 : 7. After Ryan bought another 20 marbles and Audrey gave away 80 marbles, the ratio of the number of marbles Ryan had to the number of marbles Audrey had became 1 : 3. How many marbles did Audrey have at first?
Question
The ratio of the number of marbles Ryan had to the number of marbles Audrey had at first was 2 : 7. After Ryan bought another 20 marbles and Audrey gave away 80 marbles, the ratio of the number of marbles Ryan had to the number of marbles Audrey had became 1 : 3. How many marbles did Audrey have at first?
Solution
Let's denote the number of marbles Ryan had at first as 2x and the number of marbles Audrey had at first as 7x.
According to the problem, after Ryan bought another 20 marbles and Audrey gave away 80 marbles, the ratio of the number of marbles Ryan had to the number of marbles Audrey had became 1 : 3. This can be written as:
(2x + 20) / (7x - 80) = 1/3
Cross-multiplying gives:
3*(2x + 20) = 1*(7x - 80)
Solving this equation gives:
6x + 60 = 7x - 80
This simplifies to:
x = 140
Substituting x = 140 back into the expression for the number of marbles Audrey had at first (7x) gives:
7*140 = 980
So, Audrey had 980 marbles at first.
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