Calculating Equivalent Resistance in Parallel

When resistors are connected in parallel, the current has multiple paths to flow through. The equivalent resistance of a parallel combination is always less than the smallest individual resistance in the combination.

For two resistors, \(R_1\) and \(R_2\), connected in parallel, the equivalent resistance \(R_{eq}\) is calculated using the formula:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

This formula can be rearranged to find \(R_{eq}\) directly:

\[ R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \]

In this question, we are given two resistors connected in parallel:

• \(R_1 = 10 \, \Omega\)
• \(R_2 = 20 \, \Omega\)

Let's use the formula to find the equivalent resistance.

Steps to Find Parallel Equivalent Resistance

Substitute the given values into the formula:

\[ R_{eq} = \frac{10 \, \Omega \times 20 \, \Omega}{10 \, \Omega + 20 \, \Omega} \]

First, calculate the product in the numerator:

\[ 10 \times 20 = 200 \, \Omega^2 \]

Next, calculate the sum in the denominator:

\[ 10 + 20 = 30 \, \Omega \]

Now, divide the numerator by the denominator:

\[ R_{eq} = \frac{200 \, \Omega^2}{30 \, \Omega} \]

\[ R_{eq} = \frac{200}{30} \, \Omega \]

Simplify the fraction:

\[ R_{eq} = \frac{20}{3} \, \Omega \]

Convert the fraction to a decimal:

\[ \frac{20}{3} \approx 6.6667 \, \Omega \]

Rounding the result to two decimal places, we get \(6.67 \, \Omega\).

Therefore, the equivalent resistance of a 10 Ω resistor and a 20 Ω resistor connected in parallel is approximately \(6.67 \, \Omega\).