Alright! Let's break this problem down step by step.
a. Let's first calculate the change in the equivalent resistance of the circuit. Initially, we had two 15.0 Ohm resistors, so the equivalent resistance was 15.0/2 = 7.5 Ohm. After replacing one of the resistors by a 10.0 Ohm one, the equivalent resistance (R) becomes 1 / (1/15.0 + 1/10.0) = 6 Ohm. Therefore, the change in equivalent resistance is the new resistance minus the initial one, i.e., 6-7.5 = -1.5 Ohm.
b. Now, let's focus on the change in current through the entire circuit. According to Ohm's law, the current (I) is given by the ratio of the voltage (V) to the resistance (R). Assuming a voltage of 1.5V for the circuit, the initial current was 1.5V / 7.5 Ohm = 0.2 A. After changing a resistor, the new current became 1.5V / 6 Ohm = 0.25 A. Therefore, the change in current is given by the new current minus the initial one, i.e., 0.25-0.2 = 0.05 A.
c. Finally, let's examine the change in current through one of the original 15.0 Ohm resistors after one of them was replaced. Here, we need to understand that for resistors in parallel, the voltage drop is the same across each one which implies that the current through them is defined by this voltage drop divided by the individual resistor's resistance. Therefore, the current through a 15.0 Ohm resistor, before and after the replacement of the other one, remains the same as its value doesn't change. This means that the change in individual current is 0 A.
In summary:
a) The change in equivalent resistance is -1.5 Ohm.
b) The change in current through the entire circuit is 0.05 A.
c) The change in current through one of the 15.0 Ohm resistors is 0 A.