A transmission includes a housing, an input shaft, an output shaft, a first gearset, a second gearset, a variable-ratio unit, a first torque transmitting mechanism, and a bypass torque transmitting mechanism. The first gearset is arranged between the input shaft and the output shaft, and the second gearset is coupled to the first gearset between the input shaft and the output shaft. The variable-ratio unit is configured to transmit power between (i) a first member coupled to the input shaft and the first gearset and (ii) a second member coupled to the first gearset. The first torque transmitting mechanism is selectively engageable to couple the second gearset to the housing. The bypass torque transmitting mechanism is selectively engageable to couple the first member to the second member to bypass the variable-ratio unit.