Compare the cost of dishwashing by machine versus by hand.
Both methods cost water plus the energy to heat it, and the comparison usually turns on water volume. A modern dishwasher uses around 9 to 13 litres for a full load; hand washing a comparable quantity under a running tap commonly uses 30 to 60 litres. Since heating one litre by 40 degrees takes about 0.047 kWh, that extra water is the dominant term — the energy to heat it typically outweighs the dishwasher's entire electricity draw. Hand washing only wins when you use a bowl rather than a running tap and keep the volume genuinely low, and only when the machine would otherwise run half empty.
Dishwasher cost
Cost = kWh per cycle x electricity rate + (litres / 1000) x water rate per m3
Hand washing cost
Cost = hot litres x 0.0465 kWh x electricity rate + (litres / 1000) x water rate
Because heating water is itself an electricity cost. Raising 35 litres by 40 degrees takes roughly 1.6 kWh — more than a full dishwasher cycle including its own heating, pumping and drying. Volume is the lever; whether the energy shows up on the appliance or the water heater is secondary.
Substantially. Cost per cycle is nearly fixed regardless of load, so a half-full machine roughly doubles the cost per dish. Waiting for a full load, and using the eco cycle, is what makes the machine reliably cheaper.