Problem: There are m resistor each of resistance R. First they all are connected in series and equivalent resistance is X. Now they are connected in parallel and equivalent resistance is Y. What is the ratio of X and Y?

Solution:

Understanding Series and Parallel Connections:
Resistors can be connected in two ways: series and parallel.

- Series connection: When resistors are connected in series, they are connected end to end so that the current flows through each resistor one after the other. The total resistance of the series connection is the sum of the individual resistances.
- Parallel connection: When resistors are connected in parallel, they are connected side by side so that the current can flow through each resistor simultaneously. The total resistance of the parallel connection is less than the smallest individual resistance.

Equivalent Resistance:
Equivalent resistance is the single resistance value that can replace a combination of resistors in a circuit without changing the current or voltage. It is calculated differently for series and parallel connections.

- Series equivalent resistance: The equivalent resistance of a series combination of resistors is the sum of the individual resistances. That is, R_eq = R_1 + R_2 + ... + R_n.
- Parallel equivalent resistance: The equivalent resistance of a parallel combination of resistors is the reciprocal of the sum of the reciprocals of the individual resistances. That is, 1/R_eq = 1/R_1 + 1/R_2 + ... + 1/R_n.

Ratio of X and Y:
Let's assume that there are m resistors each of resistance R. When they are connected in series, the equivalent resistance is X. Therefore, we can write:

X = mR

When the same resistors are connected in parallel, the equivalent resistance is Y. Therefore, we can write:

1/Y = 1/(mR)
Y = (mR)/m
Y = R

Therefore, the ratio of X and Y is:

X/Y = mR/R
X/Y = m

Conclusion:
The ratio of X and Y is m. This means that the equivalent resistance of a series combination of resistors is m times greater than the equivalent resistance of a parallel combination of the same resistors.