Luke had some beads. He gave ³₄ of his beads to Simon and bought 4 new beads. After that, he gave ²₇ of his remaining beads to his sister and bought another 7 new beads. In the end, Luke had 37 beads.

1. How many beads did Luke give to his sister?
2. How many beads did Luke have at first?

(a)
Fraction of  his remaining beads after he gave to his sister
= 1 - ²₇
= ⁵₇

⁵₇ of his remaining beads = 37 - 7 = 30
¹₇ of his remaining beads = 30 ÷ 5 = 6
²₇ of his remaining beads = 2 x 6 = 12

Number of beads that Luke gave to his sister = 12

(b)
⁷₇ of his remaining beads = 7 x 6 = 42

Fraction of his total beads after he gave to Simon
= 1 - ³₄
= ¹₄

¹₄ of his total beads = 42 - 4 = 38
⁴₄ of his total beads = 4 x 38 = 152

Number of beads that Luke had at first = 152 

Answer(s): (a) 12; (b) 152