## The product over sum rule.

The product over sum rule can be used as an alternative way, of finding the total resistance of two resistors connected in parallel.

## proof of product over sum rule.

• We have already seen that 1/RT = 1/R1 + 1/R2.
• Using the rules for adding fractions:
• Multiply each of the denominators to find the common denominator.
• R1x R2.
• To find the new numerators, divide the common denominator by the denominator of each fraction, then multiply by its numerator.
• ((R1x R2)/R1) x 1 = R2.
• ((R1x R2)/R2) x 1 = R1.
• Therefore 1/RT = (R2 + R1)/(R1 x R2).
• Therefore RT = (R2 x R1)/(R1 + R2).

Therefore to add two resistors in parallel, we first multiply both resistor values together, then divide by the sum of the resistor values. Note R2 x R1 is the same as R1 x R2 and so the product over sum rule is usually written as:

• RT = (R1 x R2)/(R1 + R2).

## Resistors of equal value connected in parallel.

If you have resistors of equal value connected in parallel, then to find the total resistance, you simply divide the individual resistor value by the number of resistors. i.e.

• The total resistance of:
• four 10W resistors connected in parallel is simply 10/4 = 2.5 W .
• six 18W resistors connected in parallel is simply 18/6 = 3 W .
• two 15W resistors connected in parallel is simply 15/2 = 7.5 W .
• etc.

In general, if you have "n" resistors of equal value "R, connected in parallel.

RT = R/n .

## Resistors of equal value connected in series.

If you have "n" resistors of equal value "R, connected in series.

RT = n x R .