A voltmeter of resistance `R_1` and an ammeter of resistance `R_2` are connected in series across a battery oif negligible internal resistance. When as resistance `R` is connected in parallel to voltmeter reading of ammeter increases three times white that of voltmeter reduces to one third. Find the ratio of `R_1` and `R_2``.

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the initial setup
We have a voltmeter with resistance \( R_1 \) and an ammeter with resistance \( R_2 \) connected in series across a battery with negligible internal resistance. The total voltage across the series combination is equal to the emf of the battery \( E \).

### Step 2: Write the equation for the initial condition
In the initial condition, the voltage across the ammeter \( V_A \) and the voltage across the voltmeter \( V_V \) can be expressed as:
\[
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