Section 5. The owner of any real estate upon which betterments have been assessed may, within six months after notice of such assessment has been sent out by the collector, file with the board a petition for an abatement thereof, and the board shall grant such abatement as may be necessary to make such assessment conform to section one. Such petition may be filed with the clerk or secretary of the board, or delivered by mail or otherwise at their office. The board shall within ten days after their decision upon the petition give written notice thereof to the petitioner. If an assessment is abated by the board the assessment so determined shall stand as the assessment upon the land, and if it has not been paid shall be collected in the same manner as the original assessment. If the assessment has been paid, the person by whom it was paid shall be reimbursed by the body politic on behalf of which it was assessed to the amount of the abatement allowed, with interest at the rate of six per cent per annum from the time of payment.