Dry-basis moisture is water mass divided by dry-solid mass. Initially $50=M_d(1+.13)$,so $M_d=50/1.13=44.24779$ kg. Added water is $(.35-.13)M_d=.22(44.24779)=9.73451$ kg, rounded **9.73 kg**. Treating the dry-basis percentages as wet-basis fractions would change the mass balance.
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