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Squares Ending in 55 पर समाप्त संख्याओं का वर्ग

Exact squares of two-digit numbers ending in 5, with ordinary multiplication checks.

Popularised in modern Vedic Mathematics teaching; mathematically based on an algebraic identity. Always verify with the ordinary method.

Recognition signal

The number is a positive two-digit integer that ends in 5 (15, 25, 35, 45, 55, 65, 75, 85, or 95 in this lab).

When to use

  • You need the exact square of a number that ends in 5.
  • You want a fast mental method and will verify with ordinary multiplication.

When not to use

  • The number does not end in 5.
  • The task asks for a product of two different numbers instead of one square.
  • Operands outside the current supported practice range (only the nine two-digit endings in 5).

Ordinary method

Multiply the number by itself: n × n.

Rapid method

  1. Confirm the number ends in 5.
  2. Remove the final 5 to obtain a.
  3. Calculate a × (a + 1).
  4. Append 25.
  5. Verify with ordinary multiplication.

Identity: (10a + 5)2 = 100a(a + 1) + 25

Identity: open parenthesis 10 a plus 5 close parenthesis squared equals 100 a times open parenthesis a plus 1 close parenthesis plus 25

Guided examples

Find 25²
  1. Confirm 25 ends in 5.
  2. Remove the final 5 to obtain a = 2.
  3. Calculate a × (a + 1) = 2 × 3 = 6.
  4. Append 25 → 625 = 625.
  5. Verify with ordinary multiplication: 25 × 25 = 625.

Ordinary verification: Ordinary: 25 × 25 = 625. Rapid: a = 2; 2 × 3 = 6; append 25 → 625.

Find 65²
  1. Confirm 65 ends in 5.
  2. Remove the final 5 to obtain a = 6.
  3. Calculate a × (a + 1) = 6 × 7 = 42.
  4. Append 25 → 4225 = 4225.
  5. Verify with ordinary multiplication: 65 × 65 = 4225.

Ordinary verification: Ordinary: 65 × 65 = 4225. Rapid: a = 6; 6 × 7 = 42; append 25 → 4225.

Find 95²
  1. Confirm 95 ends in 5.
  2. Remove the final 5 to obtain a = 9.
  3. Calculate a × (a + 1) = 9 × 10 = 90.
  4. Append 25 → 9025 = 9025.
  5. Verify with ordinary multiplication: 95 × 95 = 9025.

Ordinary verification: Ordinary: 95 × 95 = 9025. Rapid identity checks: a = 9; 9 × 10 = 90; append 25 → 9025.

When the rapid rule does not apply

46²

46 does not end in 5, so the ending-in-5 square shortcut does not apply. Use ordinary multiplication (or another valid method).

45 × 55

This asks for a product of two different numbers, not the square of one ending-in-5 number. The T01 square rule is for n² when n ends in 5.

Common errors

  • Error reference T01-ERR-WRONG-A.

    Forgetting to form a correctly, or using a + 1 incorrectly.

  • Error reference T01-ERR-NO-APPEND.

    Calculating a(a + 1) but forgetting to append 25.

  • Error reference T01-ERR-MISAPPLY.

    Applying the rule to a number that does not end in 5.

Verification

Always check that the rapid result equals n × n.

Direct practice

9 exact-integer questions in fixed order. Accuracy only; no timer.

Start direct practice

Basic progress summary

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