...she used 40% of the blue beads...

...the ratio of red beads to blue beads at the end was 1 : 3...

40% = 2/5

So she used 2/5 of her blue beads.

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...she used 2/5 of the blue beads...

...the ratio of red beads to blue beads at the end was 1 : 3...

Let the number of blue beads at first be 5 units.

She used 2/5 of these 5 units, or a total of 2 units of blue beads. So she had 3 units of blue beads left.

Since the ratio of red beads to blue beads in the end was 1 : 3, there must have been 1 unit of red beads at the end.

Now, we realise that if she used 45 red beads only, she would have 285 - 45 = 240 beads left.

This is for the 5 units of blue beads at first and the remaining 1 unit of red beads.

6 units = 240
1 unit = 40
4 units = 160

So there were 160 beads in the end.

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Let’s check our answers.

Ann had 285 red beads at first, made of 40 + 45 = 85 red beads and 200 blue beads.

She used 45 red beads, so she had 40 red beads left.

She used 40% of her 200 blue beads, or a total of 80 blue beads, leaving behind 120 blue beads.

Since 40 : 120 = 1 : 3 after simplification, our answers are correct.

Thank you very much!!