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Sankalana-Vyavakalanabhyam

Difference of squares multiplication trick — multiply through the midpoint

The difference of squares multiplication trick applies when two numbers sit equally far either side of a round number. Square the midpoint and subtract the square of the gap. For 47 × 53, the midpoint is 50 and the gap is 3, so the answer is 2500 − 9 = 2491 — one recalled square and one small subtraction.

This is the identity (a − b)(a + b) = a² − b² used deliberately rather than incidentally. Its value in an exam is that it converts an awkward-looking multiplication into a square you already know, which is why it pairs naturally with having squares to 30 memorised.

Try it against the clock. Difference of Squares runs at 15s easy, 20s medium and 30s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.

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How the difference of squares multiplication trick works

  1. Find the midpoint of the two numbers

    For 47 × 53 the midpoint is 50. The trick is worth using when that midpoint is a round number whose square you already know — which is the whole reason to memorise squares in the first place.

  2. Measure the gap from the midpoint to either number

    Both numbers must be the same distance away: 47 is 3 below 50 and 53 is 3 above. If the distances differ the identity does not apply and the method gives a wrong answer.

  3. Square the midpoint, subtract the square of the gap

    50² = 2500, 3² = 9, so 47 × 53 = 2491. Both values are recall rather than computation, which is what makes this fast.

  4. Use it in reverse to simplify expressions

    The same identity collapses expressions like 87² − 13² into (87 + 13)(87 − 13) = 100 × 74 = 7400. Simplification questions include this shape often, and spotting it removes two squarings.

Worked examples of the pattern

ProductBecomes
47 × 5350² − 3² = 2491
38 × 4240² − 2² = 1596
96 × 104100² − 4² = 9984
24 × 2625² − 1² = 624
87² − 13²(87+13)(87−13) = 7400
45 × 5550² − 5² = 2475

3 real difference of squares multiplication trick, with worked solutions

Generated by the same engine that mints the ComputePrep daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.

Question 1

27 × 43 = ?

Multiply via the midpoint — beat the clock.

  1. 729
  2. 1289
  3. 1121
  4. 1161
  5. 1225
Show the worked solution
  1. 27 and 43 straddle 35 by 8 — Sankalana: a × b = m² − d².
  2. 35² − 8² = 1225 − 64 = 1161.
  3. Answer: 1161.

Question 2

39 × 51 = ?

Multiply via the midpoint — beat the clock.

  1. 1521
  2. 1999
  3. 1989
  4. 2061
  5. 1986
Show the worked solution
  1. 39 and 51 straddle 45 by 6 — Sankalana: a × b = m² − d².
  2. 45² − 6² = 2025 − 36 = 1989.
  3. Answer: 1989.

Question 3

28 × 32 = ?

Multiply via the midpoint — beat the clock.

  1. 894
  2. 904
  3. 916
  4. 784
  5. 896
Show the worked solution
  1. 28 and 32 straddle 30 by 2 — Sankalana: a × b = m² − d².
  2. 30² − 2² = 900 − 4 = 896.
  3. Answer: 896.

Common mistakes this drill is built from

The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.

Using it when the gaps are unequal

47 × 54 has no common midpoint, so the identity does not hold. Check both distances before applying it — the arithmetic gives no warning when it is misused.

Adding the gap's square instead of subtracting

The identity is a² − b². Adding gives a number that is wrong by twice the small square, which is close enough to look plausible.

Not recognising the reverse form in simplification

An expression of the form x² − y² is almost always faster as (x + y)(x − y). Missing it means computing two large squares for no reason.

FAQ

When does the difference of squares trick apply?

When the two numbers are equally distant from a convenient midpoint — 47 and 53 around 50, or 96 and 104 around 100. Square the midpoint and subtract the square of the distance. If the distances are not equal, the identity does not hold.

How does it help in simplification questions?

In reverse. An expression like 87² − 13² becomes (87 + 13)(87 − 13) = 100 × 74 = 7400, which avoids squaring two awkward numbers. This shape appears regularly in simplification sets and spotting it saves most of the work.

Which squares do I need to know for this?

The round ones mostly — 20², 25², 30², 40², 50², 100². The trick's speed comes entirely from the midpoint's square being instant recall, so the memory work and the technique reinforce each other.

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