Ryan gave Sharon 2/5 of his marbles to Sharon
Sharon then gave 40% of her marbles to Ryan
Then they had equal sums of marbles in the end
Sharon had 24 more marbles than Ryan at first

For this, we will need to first convert 40% to an equivalent fraction to make this easier. 40% is 2/5, so Sharon then gave 2/5 of her marbles to Ryan.

Let 3 units be the number of marbles Sharon had at the end. Of course this was after Sharon gave away 2 units out of her 5 units of marbles (2/5 fraction), so before she gave away 2/5 of her marbles to Sharon, she had 5 units of marbles.

This also means that Ryan had 3 units of marbles at the end. And it was because Ryan received the same 2 units of marbles from Sharon, so Ryan had 1 unit of marbles before that.

So, at this point, Sharon had 5 units of marbles and Ryan had 1 unit of marbles.

Before this, Ryan gave away 2/5 of his marbles to Sharon, leaving 3/5 of his marbles behind. However, “1 unit” to represent this 3/5 fraction is unusual, so we redefine the entire story by multiplying the units by 3 from the start.

Wait for my update for the change of story.

Ryan gave Sharon 2/5 of his marbles to Sharon
Sharon then gave 40% of her marbles to Ryan (40% means 2/5)
Then they had equal sums of marbles in the end
Sharon had 24 more marbles than Ryan at first

We will approach this question in a backward manner.

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Let Sharon have 9 units of marbles at first. This would be her “3/5 of her marbles” remaining after giving Ryan marbles. So, Sharon had 15 units of marbles before this (and gave Ryan 6 units of marbles).

Similarly, Ryan had 9 units of marbles at first. This was after he received 6 units of marbles from Sharon, so he had 3 units of marbles before this.

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Sharon had 15 units of marbles at this point, while Ryan had 3 units of marbles at this point.

Ryan gave 2/5 of his marbles to Sharon, so that he would have 3/5 of his marbles left, or 3 units of marbles.

Before this, he had 5 units of marbles (and gave away 2 units of marbles to Sharon).

Sharon must have received the 2 units of marbles to obtain the 15 units of marbles, so Sharon must have had 13 units of marbles before this.

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So, at first, Sharon had 13 units of marbles while Ryan had 5 units of marbles. You can check that they will both have 9 units of marbles at the end after the series of exchanges.

Therefore, Sharon had 8 units of marbles more than Ryan at first, and this was represented by 24 marbles.

8 units = 24 marbles
1 unit = 3 marbles

Recall that Ryan had 5 units of marbles at first.

5 units = 15 marbles

So, Ryan had 15 marbles at first.