I understand that for the ideal op amp, the voltages at both input nodes are equal to each other. However, because each of the voltages are divided across varying resistances, I do not understand how the 2 current values are equal. Can someone please help me understand this?
It's just a case of asking yourself what current flows into the input of an ideal op-amp then seeing what remains to flow in the resistors: -
The trouble is that it looks like the set answer is wrong because the current into the 1 kΩ resistor is upwards and not downwards. So, the right answer should be \$-i_0\$ as I see it.