The total resistance of a circuit in series can be described by the equation:

The series circuit in ths problem therefore has a resistance of:

The resistance of a circuit wired in parallel has a total resistance of:

We are assuming that the two circuits have the same total resistance, so to find the resistance of the unknown resistor, we set up the following equation:

This, when solved, gives us a resitance of 200 ohms for our unknown element.

The equivalent voltage of batteries connected in series is the sum of the voltage of each battery, or

In our problem,

The equivalent voltage of batteries connected in parallel is equal to the voltage of 1 battery.

In this problem, 

Note: Connecting batteries in parallel increases the capacity of the battery.

To find the equivalent resistance of this system, we must first find the equivalent resistance of the resistors in parallel, then evaluate the resistors in parallel.

The parallel resistor equivalence is given by the following equation,

In our problem,

The parallel resistors can now be treated as one resistor with the resistance .  To find the total resistance, we add the resistances of ,  and .

To find the equivalent resistance of this system, we must first find the equivalent resistance of the resistors in parallel, then evaluate the resistors in parallel.

The parallel resistor equivalence is given by the following equation,

In our problem,

The parallel resistors can now be treated as one resistor with the resistance .  To find the total resistance, we add the resistance of  and .

Let's go through and figure out what the equivalent resistances are.

(A) This is a resistor in parallel with two series resistors. This looks like:

(B) These are just two resistors in series,

(C) This is just one resistor,

(D) All three resistors are in parallel,

Ranking them we see that the largest to smallest values are 

The key hallmark of parallel circuits is that the elements connected in parallel have the same voltage drop across them. It doesn’t matter if the circuit element is a resistor, capacitor, or an inductor; the voltage drop across all elements is the same. This means that the voltage across both resistors A and B is same. On the other hand, circuit elements connected in series have the same current flowing through them; however, they have different voltage drops.

We need to use the principles of circuits and Ohm’s law to solve this question. Recall that circuit elements (in this question resistors) connected in series have the same current flowing through them. The current, therefore, flowing through all three resistors is . To calculate the voltage we need to first calculate an equivalent resistance of the circuit (a single resistor that models the three resistors). Since the resistors are connected in series, we can simply add the resistance of each resistor to get the equivalent resistance, .

Using Ohm’s law we can now calculate the voltage provided.

To solve this question we need to understand the principles of circuits. Circuit elements (such as resistors, capacitors, and inductors) connected in parallel have the same voltage whereas circuit elements connected in series have the same current flowing through them. In parallel circuits, the current hits nodes (regions where two circuit elements branch out and become parallel) and gets split into two different currents, each supplying the individual circuit elements; therefore, the current flowing through parallel connected circuit elements is always less than the total flow of the current. On the other hand, circuit elements in series have the same current flowing through them; therefore, the total current of the circuit flows through each element connected in series. This means that the current flowing through series circuit is higher.

Voltage drop across the parallel circuit elements is equal; therefore, the voltage equals the voltage supplied by the power source. In circuit elements in series, however, the voltage drop is smaller across each element; therefore, resistors in series have smaller voltage drop and larger current than their parallel circuit counterparts.

The amount of brightness that each light bulb will produce is proportional to the amount of current passing though it. If the lightbulbs were to be placed in series, the total amount of current passing though each bulb would be equal to the voltage of the battery divided by the sum of the resistances of both lightbulbs, given by:

If the lightbulbs were instead to be placed in parallel, the total amount of current passing though would be equal to the voltage across the lightbulb, equal to the voltage of the battery, divided by the internal resistance of the lightbulb:

From those two equations, it is clear to see that 

, and that there would be more light if the lightbulbs are placed parallel to one another.