Natalie and Damien had the same number of marbles at first. Natalie gave 23 marbles to her friend. Damien's friend gave him 11 marbles. The number of marbles Damien had in the end was three times as many as the number of marbles Natalie had. How many marbles did each of them have at first?
No x + x = 23
Let's assume that the initial number of marbles both Natalie and Damien had is 'x'.
After Natalie gave 23 marbles to her friend, she was left with x - 23 marbles.
Similarly, after Damien's friend gave him 11 marbles, he had x + 11 marbles.
According to the given information, the number of marbles Damien had in the end (x + 11) was three times as many as the number of marbles Natalie had in the end (x - 23).
Therefore, we can set up the equation:
x + 11 = 3(x - 23)
Expanding the equation:
x + 11 = 3x - 69
Rearranging the equation to isolate x:
2x = 80
x = 40
Therefore, Natalie and Damien had 40 marbles each at first.
To solve this problem, let's assign variables to the number of marbles Natalie and Damien had at first. Let's say Natalie had "x" marbles and Damien had "x" marbles as well.
According to the given information, Natalie gave away 23 marbles, so the number of marbles she had in the end is (x - 23). Damien, on the other hand, received 11 marbles, so the number of marbles he had in the end is (x + 11).
Now, we are told that Damien had three times as many marbles as Natalie had. So, we can set up the equation:
x + 11 = 3(x - 23)
Let's solve this equation step by step:
x + 11 = 3x - 69 (Distribute the 3 on the right side)
11 + 69 = 3x - x (Combine like terms)
80 = 2x (Simplify the equation)
40 = x (Divide both sides by 2)
Therefore, Natalie had 40 marbles at first. Since Damien also had the same number of marbles initially, Damien also had 40 marbles at first.
at the beginning:
Natalie : x
Damien : x
Natalie gave away 23, so she now has x - 23
Damien got 11 marbles, so now he has x + 11
Damien has 3 times as many as Natalie
x+11 = 3(x-23)
solve for x