Resistors in Parallel: Understanding the Problem

The problem describes three resistors, R1, R2, and R3, whose resistance values are in a specific ratio of 2 : 3 : 4. These resistors are connected in a parallel configuration. The combined effect of these parallel resistors is given as an equivalent resistance of 24 Ω. Our goal is to determine the individual resistance values of R1, R2, and R3.

Resistance Ratio: Setting up the Expressions

Given the ratio of the resistances R1 : R2 : R3 = 2 : 3 : 4, we can express each individual resistance in terms of a common proportionality constant, let's call it 'x'. This method ensures that the ratio among the resistances is maintained while allowing us to calculate their exact values.

• R1 can be written as \(2x \text{ Ω}\)
• R2 can be written as \(3x \text{ Ω}\)
• R3 can be written as \(4x \text{ Ω}\)

Equivalent Resistance in Parallel Circuits

For resistors connected in parallel, the reciprocal of the equivalent resistance (\(R_{eq}\)) is equal to the sum of the reciprocals of the individual resistances. This relationship is fundamental for analyzing parallel circuits. The general formula for three resistors in parallel is:

$$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $$

We are provided with the equivalent resistance \(R_{eq} = 24 \text{ Ω}\). Now, we substitute this value and the expressions for R1, R2, and R3 (in terms of x) into the parallel resistance formula:

$$ \frac{1}{24} = \frac{1}{2x} + \frac{1}{3x} + \frac{1}{4x} $$

Calculating the Common Factor 'x'

To solve for 'x', we need to combine the fractions on the right-hand side of the equation. We find the least common multiple (LCM) of the denominators \(2x\), \(3x\), and \(4x\). The LCM of 2, 3, and 4 is 12, so the common denominator for the terms involving 'x' will be \(12x\).

$$ \frac{1}{24} = \frac{1 \times 6}{2x \times 6} + \frac{1 \times 4}{3x \times 4} + \frac{1 \times 3}{4x \times 3} $$

$$ \frac{1}{24} = \frac{6}{12x} + \frac{4}{12x} + \frac{3}{12x} $$

Now, add the numerators while keeping the common denominator:

$$ \frac{1}{24} = \frac{6 + 4 + 3}{12x} $$

$$ \frac{1}{24} = \frac{13}{12x} $$

To isolate 'x', we can cross-multiply:

$$ 1 \times 12x = 13 \times 24 $$

$$ 12x = 312 $$

Finally, divide both sides by 12 to find the value of 'x':

$$ x = \frac{312}{12} $$

$$ x = 26 $$

Thus, the proportionality constant 'x' is 26.

Individual Resistances: Final Values

With the value of 'x' determined, we can now calculate the individual resistance for each resistor, R1, R2, and R3, using the expressions we set up earlier:

• R1 = \(2x = 2 \times 26 = 52 \text{ Ω}\)
• R2 = \(3x = 3 \times 26 = 78 \text{ Ω}\)
• R3 = \(4x = 4 \times 26 = 104 \text{ Ω}\)

Therefore, the individual resistances are 52 Ω, 78 Ω, and 104 Ω.

Summary of Resistance Values