Question

When two resistors are connected in series, their effective resistance is 80 Ω. When they are connected in parallel, their effective resistance is 20 Ω. What are the values of the two resistances?

Answer 1

Given:

Effective resistors in Series, R_S = 80 Ω

Effective resistors in Series, R_P = 20 Ω

Find: resistors (R₁ and R₂)

Formula:

1. R_S = R₁ + R₂
2. `1/"R"_"P" = 1/"R"_1 + 1/"R"_2`

Calculations:

From Formula (1),

80 = R₁ + R₂

From Formula (2),

`1/20 = 1/"R"_1 + 1/"R"_2`

`1/20 = ("R"_2 + "R"_1)/("R"_1 xx "R"_2)`

∵ R₁ + R₂ = 80

∴ `1/20 = 80/("R"_1 xx "R"_2)`

∴ R₁ × R₂ = 1600

∴ R₁ = `1600/"R"_2`

Substituting this value in formula (i),

`1600/"R"_2 + "R"_2 = 80`

∴ `1600 + "R"_2^2 = 80  "R"_2`

∴ `"R"_2^2 - 80"R"_2 + 1600 = 0`

∴ `"R"_2^2 - 40"R"_2 - 40"R"_2 + 1600 = 0`

∴ `"R"_2("R"_2 - 40) - 40("R"_2 - 40) = 0`

(R₂ − 40) (R₂ − 40) = 0

∴ R₂ = 40 Ω

∴ R₁ + 40 = 80

∴ R₁ = 40 Ω

The values of two resistors are 40 Ω and 40 Ω.