Let's set up the equations to solve the problem step by step:
Let's assume Rani had x green beads at first. Then she had (x + 216) blue beads.
After buying additional beads:
The number of blue beads increased by 25%, which means she bought 0.25 * (x + 216) more blue beads.
The number of green beads increased by 50%, which means she bought 0.5 * x more green beads.
According to the given information, she bought 14 more blue beads than green beads. So, we can set up the equation:
0.25 * (x + 216) = 0.5 * x + 14
Now, let's solve for x:
0.25x + 54 = 0.5x + 14
0.5x - 0.25x = 54 - 14
0.25x = 40
x = 40 / 0.25
x = 160
So, Rani had 160 green beads at first.
Now, to find the number of blue beads at the end, we need to add 216 (initial difference) + 25% increase:
Number of blue beads at the end = (x + 216) + 0.25 * (x + 216)
Number of blue beads at the end = 160 + 216 + 0.25 * (160 + 216)
Number of blue beads at the end = 160 + 216 + 0.25 * 376
Number of blue beads at the end = 160 + 216 + 94
Number of blue beads at the end = 470
So, Rani would have 470 blue beads in the end.