Paper I · 9 Logical Reasoning & Analytical Ability · 9.1 Number system and simplification

9.1.13 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 Paper I · Paper I
  • Standard level
  • 5 min read
  • Free

Free PDF

Download this PDF

Enter your email once and the PDF is yours. We send exam updates and free material; you can stop them any time.

In the print window, choose Save as PDF.

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 Paper I 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.

Practise this topic. Questions on Numerical ability, with answers explained.

Practise in the app Logical Reasoning & Analytical Ability questions with answers

All Number system and simplification notes · APPSC Paper I syllabus · APPSC Paper I Mock 1 with answers · Start free

Medha AI writes its own notes and checks them against the sources listed. Medha AI is an independent preparation service. It is not affiliated with, endorsed by or connected to APPSC, TSPSC, MPSC, UPSC or any other commission or government body.