Question
John and Peter had 96 stamps altogether. After John had given Peter 1/2 of his stamps, Peter then gave John 1/5 of his stamps. In the end, both of them had the same number of stamps each. How many stamps did Peter have at first?
Answers
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Let's start by setting up some equations to represent the given information.
Let J be the number of stamps John had at first, and let P be the number of stamps Peter had at first.
We know that John and Peter had 96 stamps altogether, so we can write:
J + P = 96
We also know that after John gave Peter 1/2 of his stamps, he was left with only 1/2 of his original amount. That means John gave away 1/2 of his stamps, or (1/2)J stamps. So John's new number of stamps is:
J - (1/2)J = (1/2)J
Similarly, after Peter gave John 1/5 of his stamps, he was left with 4/5 of his original amount. That means Peter gave away 1/5 of his stamps, or (1/5)P stamps. So Peter's new number of stamps is:
P - (1/5)P = (4/5)P
Finally, we are told that after these exchanges, John and Peter had the same number of stamps each. That means:
(1/2)J = (4/5)P
We can use this last equation to solve for one of the variables in terms of the other. For example, we could solve for J:
J = (4/5)(2/1)P
J = (8/5)P
Now we can substitute this expression for J into the first equation we wrote:
(8/5)P + P = 96
Simplifying this equation, we get:
(13/5)P = 96
Multiplying both sides by 5/13 gives us:
P = 360/13
This is approximately equal to 27.69, so we can conclude that Peter had about 27 stamps at first.
Let J be the number of stamps John had at first, and let P be the number of stamps Peter had at first.
We know that John and Peter had 96 stamps altogether, so we can write:
J + P = 96
We also know that after John gave Peter 1/2 of his stamps, he was left with only 1/2 of his original amount. That means John gave away 1/2 of his stamps, or (1/2)J stamps. So John's new number of stamps is:
J - (1/2)J = (1/2)J
Similarly, after Peter gave John 1/5 of his stamps, he was left with 4/5 of his original amount. That means Peter gave away 1/5 of his stamps, or (1/5)P stamps. So Peter's new number of stamps is:
P - (1/5)P = (4/5)P
Finally, we are told that after these exchanges, John and Peter had the same number of stamps each. That means:
(1/2)J = (4/5)P
We can use this last equation to solve for one of the variables in terms of the other. For example, we could solve for J:
J = (4/5)(2/1)P
J = (8/5)P
Now we can substitute this expression for J into the first equation we wrote:
(8/5)P + P = 96
Simplifying this equation, we get:
(13/5)P = 96
Multiplying both sides by 5/13 gives us:
P = 360/13
This is approximately equal to 27.69, so we can conclude that Peter had about 27 stamps at first.