Ivan had some beads. He gave ³₄ of his beads to Daniel and bought 4 new beads. After that, he gave ²₅ of his remaining beads to his cousin and bought another 7 new beads. In the end, Ivan had 31 beads.

1. How many beads did Ivan give to his cousin?
2. How many beads did Ivan have at first?

(a)
Fraction of  his remaining beads after he gave to his cousin
= 1 - ²₅
= ³₅

³₅ of his remaining beads = 31 - 7 = 24
¹₅ of his remaining beads = 24 ÷ 3 = 8
²₅ of his remaining beads = 2 x 8 = 16

Number of beads that Ivan gave to his cousin = 16

(b)
⁵₅ of his remaining beads = 5 x 8 = 40

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

¹₄ of his total beads = 40 - 4 = 36
⁴₄ of his total beads = 4 x 36 = 144

Number of beads that Ivan had at first = 144 

Answer(s): (a) 16; (b) 144