At FIRST,a bag contained 9 red beads and some blur beads. After some red beads were added to the bag,2/5 of the beads were blue.then,56yellow beads were added to the bag and now,2/9 of the beads were blue .How many red beads were added to the bag?

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Let's solve this step by step. We have a bag that initially contains 9 red beads and some unknown number of blue beads. Let's call the initial number of blue beads B, and the number of red beads added R. We need to find the value of R. Now let's set up our first equation. After adding R red beads to the bag, we have a total of 9 plus B plus R beads. The number of blue beads remains B. We're told that 2 fifths of all beads are blue, so B divided by 9 plus B plus R equals 2 fifths. Cross multiplying gives us 5B equals 2 times the quantity 9 plus B plus R. Expanding this gives 5B equals 18 plus 2B plus 2R. Simplifying, we get 3B equals 18 plus 2R, which is our first equation. Now we add 56 yellow beads to the bag. The total number of beads becomes 9 plus B plus R plus 56. The number of blue beads is still B, since we only added yellow beads. We're told that now 2 ninths of all beads are blue. So B divided by 9 plus B plus R plus 56 equals 2 ninths. Cross multiplying gives 9B equals 2 times the quantity 9 plus B plus R plus 56. This simplifies to 9B equals 18 plus 2B plus 2R plus 112, which equals 2B plus 2R plus 130. Subtracting 2B from both sides gives us 7B equals 2R plus 130, our second equation. Now we solve the system of two equations. From equation 1, we can express 2R as 3B minus 18. Substituting this into equation 2 gives us 7B equals 3B minus 18 plus 130, which simplifies to 7B equals 3B plus 112. Subtracting 3B from both sides gives 4B equals 112, so B equals 28. Now we substitute B equals 28 back into equation 1: 3 times 28 equals 18 plus 2R, which gives us 84 equals 18 plus 2R. Solving for R, we get 66 equals 2R, so R equals 33. Let's verify our answer. With B equals 28 and R equals 33, after adding 33 red beads, we have a total of 9 plus 28 plus 33 equals 70 beads. The blue fraction is 28 over 70, which equals 2 fifths. This matches our first condition. After adding 56 yellow beads, we have 70 plus 56 equals 126 total beads. The blue fraction is now 28 over 126, which equals 2 ninths. This matches our second condition. Therefore, our answer is correct: 33 red beads were added to the bag.