Valen had some coins. He gave ³₄ of his coins to Xavier and bought 4 new coins. After that, he gave ²₅ of his remaining coins to his sister and bought another 7 new coins. In the end, Valen had 34 coins.

1. How many coins did Valen give to his sister?
2. How many coins did Valen have at first?

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

³₅ of his remaining coins = 34 - 7 = 27
¹₅ of his remaining coins = 27 ÷ 3 = 9
²₅ of his remaining coins = 2 x 9 = 18

Number of coins that Valen gave to his sister = 18

(b)
⁵₅ of his remaining coins = 5 x 9 = 45

Fraction of his total coins after he gave to Xavier
= 1 - ³₄
= ¹₄

¹₄ of his total coins = 45 - 4 = 41
⁴₄ of his total coins = 4 x 41 = 164

Number of coins that Valen had at first = 164 

Answer(s): (a) 18; (b) 164