A owes B Rs 1000/- but the debt is barred by the Limitation Act. A sign a written promise to pay B Rs 500 on account of debt:

Such an agreement is a contract as per S.25(3) of Indian Contract Act, 1872.   According to Sec 25 “An Agreement without consideration is void unless- (1)        it is expressed in writing and registered under the law for the time being in force for the registration of documents, and is made on account of natural love and affection between parties standing in a near relation to each other; or unless (2)        it is a promise to compensate, wholly or in part, a person who has already voluntarily done something for the promisor, or something which the promisor was legally compellable to do; or unless. (3)        It is a promise, made in writing and signed by the person to be charged therewith, or by his agent generally or specially authorized in that behalf, to pay wholly or in part a debt of which the creditor might have enforced payment but for the law for the limitation of suits.   In any of these cases, such an agreement is a contract.   Illustrations   (e) A owes B Rs. 1,000, but the debt is barred by the Limitation Act. A signs a written promise to pay B Rs. 500 on account of the debt. This is a contract."