Tony and Peter both had some marbles at first. After Tony gave Peter 4/14 of his marbles, the ratio of Tony's marbles to Peter's marbles became 5:3. What was the ratio of Peter's marbles to Tony's marbles at first?
Let's assume that Tony had T marbles at first and Peter had P marbles at first.
After Tony gave Peter 4/14 of his marbles, Tony had T - (4/14)T = (10/14)T marbles left.
The ratio of Tony's marbles to Peter's marbles became 5:3, so (10/14)T / P = 5/3
Cross multiplying, we get (10/14)T = (5/3)P
Simplifying, we get (5/7)T = (5/3)P
Dividing both sides by (5/3), we get T = (7/3)P
Therefore, the ratio of Peter's marbles to Tony's marbles at first was P/T = P / ((7/3)P) = 3/7. Answer: \boxed{3:7}.
Let’s assume that Tony had x marbles and Peter had y marbles at first.
According to the given information:
After Tony gave Peter 4/14 of his marbles, Tony had x - (4/14)x = (10/14)x marbles and Peter had y + (4/14)x marbles.
The ratio of Tony's marbles to Peter's marbles became 5:3, which means:
(10/14)x / (y + (4/14)x) = 5/3
To simplify the equation, we can cross multiply:
3 * (10/14)x = 5 * (y + (4/14)x)
Simplifying further:
(30/14)x = 5y + (20/14)x
Multiplying both sides by 14 to eliminate fractions:
30x = 70y + 20x
Combining like terms:
10x = 70y
Dividing both sides by 10:
x = 7y
Therefore, the ratio of Peter's marbles to Tony's marbles at first was 7:1.