Prelims · 8 Mental Ability · 8.1 Number system and simplification

8.1.13 Speed maths: multiplication, squaring and percentage shortcuts

Most aptitude arithmetic is a handful of patterns: turn 5, 25 and 125 into divisions, 9s into subtractions, nearby numbers into a base, and percentages into fractions.

  • APPSC Group 2 · Prelims
  • Standard level
  • 5 min read
  • Free

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Core facts

  • × 5 = × 10 ÷ 2; × 25 = × 100 ÷ 4; × 125 = × 1,000 ÷ 8.
  • × 9, 99, 999: N × 999 = N × 1,000 − N.
  • × 11 (two-digit ab): write a, (a + b), b, carrying if a + b ≥ 10: 63 × 11 = 693; 78 × 11 = 7(15)8 = 858.
  • Square of a number ending in 5: n5² = n × (n + 1), then write 25: 85² = 72|25.
  • Same tens digit, units adding to 10: 64 × 66 = (6 × 7)|(4 × 6) = 4,224.
  • Difference of squares: (a + b)(a − b) = a² − b²: 73 × 67 = 70² − 3².
  • Fraction–percent table:
FractionPercentFractionPercent
1/616.67%1/119.09%
1/714.29%1/128.33%
1/812.5%1/166.25%
1/911.11%3/837.5%
  • x% of y = y% of x.

Example 1. Find 4,628 × 25. Solution. 4,62,800 ÷ 4. Answer: 1,15,700

Example 2. Find 85². Solution. 8 × 9 = 72, then 25. Answer: 7,225

Example 3. Find 78 × 11. Solution. 7, (7 + 8 = 15), 8: carry 1 to get 858. Answer: 858

Explained

Type 1: multiplying by 9s.

Example 4. Find 7,346 × 999. Solution. 73,46,000 − 7,346. Answer: 73,38,654

Type 2: squaring near a base. (Base ± d)² = base² ± 2 × base × d + d². For base 100: (100 − d)² is (100 − 2d) followed by d² as two digits.

Example 5. Find (i) 97², (ii) 104², (iii) 996². Solution. (i) 97 is 3 below 100: 94|09. (ii) 104 is 4 above: 108|16. (iii) Base 1,000, 4 below: 992|016. Answer: (i) 9,409 (ii) 10,816 (iii) 9,92,016

Type 3: same tens digit, units summing to 10.

Example 6. Find 64 × 66. Solution. 6 × 7 = 42; 4 × 6 = 24. Answer: 4,224

Type 4: difference of squares. Use it when the two numbers are equally spaced around a round number.

Example 7. Find 73 × 67. (a) 4,881 (b) 4,891 (c) 4,901 (d) 4,991 Solution. 70² − 3² = 4,900 − 9. Answer: (b) 4,891

Type 5: percentage shortcuts.

Example 8. Find (i) 18% of 450, (ii) 12.5% of 968, (iii) 16⅔% of 4,326. Solution. (i) Swap: 18% of 450 = 45% of 180 = 90 − 9 = 81. (ii) 1/8 of 968 = 121. (iii) 1/6 of 4,326 = 721. Answer: (i) 81 (ii) 121 (iii) 721

Example 9. Find 4,321 × 125. Solution. 43,21,000 ÷ 8 = 5,40,125. Answer: 5,40,125

Example 10. Find 46 × 25 × 4. Solution. Pair 25 × 4 = 100 first. Answer: 4,600

Speed tips.

  • Rearrange products to pair 2 × 5, 4 × 25 or 8 × 125 before multiplying.
  • Write Indian commas (lakhs, crores) as you go; misplaced zeros cause most errors.

Common traps

  • ✗ "× 25 means × 100 × 4." ✓ It is × 100 ÷ 4.
  • ✗ "97² = 94|9 = 949." ✓ The right part needs two digits for base 100: 94|09 = 9,409.
  • ✗ "+20% then −20% leaves the price unchanged." ✓ Net change is −4%.
  • ✗ "78 × 11 = 7158." ✓ Carry the 1 from 15: 858.
  • ✗ "Digit roots match, so the product is right." ✓ Also check the last two digits; digit roots miss swapped digits.

Quick revision

  • × 5: ÷ 2 with a 0; × 25: ÷ 4 with 00; × 125: ÷ 8 with 000.
  • N × 99…9 = N × 10ᵏ − N.
  • ab × 11 = a | a + b | b.
  • n5² = n(n + 1) | 25.
  • Same tens, units adding to 10: a(a + 1) | product of units.
  • (100 ± d)² = (100 ± 2d) | d².
  • (a + b)(a − b) = a² − b².
  • 1/6 = 16.67%, 1/7 = 14.29%, 1/8 = 12.5%, 1/12 = 8.33%.
  • Successive change: a + b + ab/100.
  • Check products by digit root and the last two digits.
Go deeper: Deeper (beyond what APPSC Group 2 usually asks)

Deeper Advanced

Example 11. Find 108 × 96 by the base method. Solution. Against base 100, the numbers are +8 and −4. Cross-add: 108 − 4 = 104, so 104 hundreds. Multiply the differences: 8 × (−4) = −32. 10,400 − 32. Answer: 10,368

Example 12. Find 1,999 × 2,001 and 412². Solution. 1,999 × 2,001 = 2,000² − 1 = 39,99,999. 412² = 400² + 2 × 400 × 12 + 12² = 1,60,000 + 9,600 + 144 = 1,69,744. Answer: 39,99,999 and 1,69,744

Example 13. A price rises by 20% and then falls by 20%. What is the net change? (a) 0% (b) −2% (c) −4% (d) +4% Solution. Net % = a + b + ab/100 = 20 − 20 − 400/100 = −4. (Check: 1.2 × 0.8 = 0.96.) Answer: (c) −4%

Example 14. Without full multiplication, which is 2,347 × 386? (a) 9,05,942 (b) 9,05,492 (c) 9,04,942 (d) 9,15,942 Solution. Casting out nines: 2,347 → digit root 7; 386 → 8; 7 × 8 = 56 → 2. Digit roots of the options: (a) 2, (b) 2, (c) 1, (d) 3. Both (a) and (b) end in 2, so check the last two digits: 47 × 86 = 4,042 ends in 42, which removes (b) (…92). Answer: (a) 9,05,942

  • The digit-root check catches most errors but not swapped digits; combine it with the last one or two digits.

Example 15. Find 989 × 1,011. Solution. 1,000² − 11² = 10,00,000 − 121. Answer: 9,99,879

Sources: Standard algebraic identities ((a + b)(a − b), (a ± b)², base-method multiplication); NCERT Mathematics textbooks, Classes 7–8 (percentages, squares)

Last checked 2026-10-17.

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