What is the ruling of the Sharia regarding an agreement on a worker's wage if there was a misunderstanding of the wage amount between the two parties? Is it permissible to pay the remaining amount on her behalf without her knowledge? And is she considered sinful?

If the renter and the hired person disagree on the amount of the wage after the benefit has been consumed, without evidence: - The Shafi'is and Hanbalis are of the view that both parties should take an oath (tahāluf). If both swear, the hired person is entitled to the customary wage (ujrat al-mithl), estimated by experts. If one party refuses to swear, judgment is given in favor of the other. - The Hanafis are of the view that they do not take an oath, and the word of the renter is accepted with his oath, because he is denying the increase, and the default is its absence. - Ibn Qudamah is of the view that the tahāluf occurs when there is a disagreement on the amount of the wage. If the contract is annulled after the period, the customary wage (ujrat al-mithl) becomes due. - Al-Zayla'i is of the view that the word of the renter is accepted with his oath after the benefits have been consumed.

Your volunteering to resolve the dispute is a good deed, and your grandmother does not need to know about it.