Denominator for each type of bead represents the number of units there are for each type of bead.

From the question info we can tell that the ratio of black to white beads is 1 : 3

So we should make the denominator for each fraction such that it fulfills this ratio.

Fraction of black beads used = ¼ = 2/8

Fraction of white beads used = 3/8 = 9/24

So we can say that there are 8 units of black beads at first and 24 units of white beads at first. A total of 32 units.

Number of units of beads used = 9 + 2 = 11

Fraction of beads used = 11/32

Number of units left = 32 - 11 = 21

21 units = 63

Number of beads at first = 63 x 32/21
= 96

Thank you J for the clear explanation

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