Suppose there is a circuit consister of only resistance and batteries , suppose one is to double (or increase it to in n-times )all voltage and all resistances , show that currents are unalatered

Explanation

Let the effective internal resistance of the battery is `R_(eff)`, the effective exeternal resistance R and the effective voltage of the battery is `V_(eff)`.
Applying Ohm's law.
then current through R is given by
`I=(V_(eff))/(R_(eff)+R)`
If all the resistances and the effective voltage are increased n-times, then we have
`V_(eff)^("new")=nV_(eff),R_(eff)^("new")=nR_(eff)`
The, the new current is given by
`I'=(nV_(eff))/(nR_(eff)+nR)=(n(V_(eff)))/(n(R_(eff)+R))=((V_(eff)))/((R_(eff)+R))=I`