NDA (II) 2016 General Ability Test · Question 113 of 150

Correct — C, 9x. Work it in two short steps. In parallel, the reciprocals add, so 1/x = 1/r + 1/r + 1/r = 3/r, which gives x = r/3 and therefore r = 3x. In series the resistances simply add, so the total is r + r + r = 3r. Substituting r = 3x into that gives 3 × 3x = 9x. The general result worth carrying away is that for n identical resistors the series total is n² times the parallel total, and with n = 3 that factor is nine.

• (a)3x — The most-marked wrong option, and it is exactly one step short. The quantity 3x is r, the resistance of a single one of the three resistors — true, but the question asks for the series total, which is three of them added, not one.
• (b)3rx — Multiplying two resistances gives something with the units of ohm squared, so this cannot be a resistance at all. A quick dimensional glance rules it out before any algebra is needed.
• (d)3/x — Dividing a pure number by a resistance gives the units of conductance, the reciprocal of resistance, not resistance. It also moves the wrong way: making each resistor larger would make this expression smaller, which is the opposite of what a series combination does.

Resistances in series carry the same current and their potential differences add, so the equivalent resistance is the plain sum, R = R₁ + R₂ + R₃. Resistances in parallel share the same potential difference and their currents add, so it is the reciprocals that sum, 1/R = 1/R₁ + 1/R₂ + 1/R₃. A series combination is therefore always larger than the largest single resistor in it, and a parallel combination always smaller than the smallest.

The exam sets this pattern year after year with different numbers, so it pays to hold the identical-resistor shortcuts ready: n equal resistors of value r give nr in series and r/n in parallel, and the ratio between the two is n². The other habit that saves marks is a units check on every option before you commit — in a question whose answer must be a resistance, any option that multiplies or divides two resistances can be discarded on sight, which here removes options (b) and (d) immediately and leaves a straight choice between 3x and 9x.

• Resistances in series add directly; resistances in parallel add as reciprocals.
• For n identical resistors of resistance r, the series total is nr and the parallel total is r/n.
• The series equivalent is n² times the parallel equivalent for n identical resistors — here nine times.
• A parallel combination is always smaller than the smallest resistor in it, which is why household appliances are wired in parallel across the supply.

• Stopping at r = 3x and marking option (a) — it is the value of one resistor, not of the three in series.
• Forgetting that it is the reciprocals, not the resistances, that add in parallel.
• Skipping the units check, which would have removed two options for free.

Combination-of-resistors items are almost guaranteed in the GAT, sometimes as a numerical with values and sometimes, as here, purely in symbols, so practise both forms.

• practice — not a real PYQ

  Four resistors, each of 8 ohm, are connected in parallel. The equivalent resistance of the combination is

  • (a)2 ohm
  • (b)4 ohm
  • (c)16 ohm
  • (d)32 ohm

  Answer(a) 2 ohm — for n identical resistors in parallel the equivalent is r/n, that is 8/4.

• practice — not a real PYQ

  Two resistors give an equivalent resistance of 6 ohm in series and 1.5 ohm in parallel had they been equal to each other. Each resistor is therefore

  • (a)1.5 ohm
  • (b)3 ohm
  • (c)6 ohm
  • (d)12 ohm

  Answer(b) 3 ohm — two equal resistors give 2r in series and r/2 in parallel, and both 2r = 6 and r/2 = 1.5 give r = 3 ohm.