Question

Two resistances r1 and r2 give combined resistance of 4.5 ohms when in series and 1 ohm when in parallel. the resistances are

Answer 1

r1+r2=4.5
r3=1
r1+r2 is parallel to r3
so
1/rp=1/4.5+1/1
1/rp= (1+4.5)/4.5
1/rp=5.5/4.5
rp=4.5/5.5

Answer 2

The value of resistance $\bold{R_{1} = 3}$ ohm and $\bold{R_{2}= 1.5}$ ohm.

Solution:

Given: $R_{1}+R_{2}=4.5$ as connected in series.

The given resistance have different equivalent resistance when placed in two different arrangement, thus the resistances be –

And, $\bold{\frac{R_{1} \times R_{2}}{R_{1}+R_{2}}=1}$ as connected in parallel.

As known $R_{1}+R_{2}=4.5$  

$\begin{array}{l}{R_{1} \times R_{2}=4.5 \times 1} \\ \\{R_{1} \times R_{2}=4.5} \\ \\{R_{1}=\frac{4.5}{R_{2}}}\end{array}$

Place the value of  $R_{1}$ into series connection equation, we get,

$\begin{array}{l}{\left(\frac{4.5}{R_{2}}\right)+R_{2}=4.5} \\ {R_{2}^{2}-4.5 R_{2}+4.5=0}\end{array}$

On solving we get, $R_{2}=1.5$

$R_{1}=4.5-1.5=3$

Thereby, $\bold{R_{1}=3}$ ohm and $\bold{R_{2}=1.5}$ ohm