Chapter 9
📊 Percentage Mastery
🔥 Why Percentages Matter
Percentages appear in every competitive exam — SSC, Banking, JEE, UPSC. Master these Vedic tricks and you'll solve percentage problems 3× faster than the conventional method!
In this chapter, you'll learn to calculate ANY percentage mentally — no pen, no paper, no calculator.
"The mind is the best calculator. Train it with the right techniques, and numbers will dance before you." — Vedic Mathematics Principle
The 10% Foundation Trick
Every percentage calculation starts with 10%. To find 10% of any number, simply move the decimal point one place to the left.
10% = Move decimal one place LEFT
10% of 450 = 45.0
10% of 1,230 = 123.0
10% of 87 = 8.7
10% of 3.6 = 0.36
This is the foundation of ALL percentage shortcuts!
10% of 450 = 45.0
10% of 1,230 = 123.0
10% of 87 = 8.7
10% of 3.6 = 0.36
This is the foundation of ALL percentage shortcuts!
Find 10% of 6,750
Move decimal one place left: 6,750 → 675.0
Answer: 675
The 5% Trick — Half of 10%
5% = Half of 10%
5% of 450 → 10% = 45 → half = 22.5
5% of 840 → 10% = 84 → half = 42
5% of 1,600 → 10% = 160 → half = 80
5% of 450 → 10% = 45 → half = 22.5
5% of 840 → 10% = 84 → half = 42
5% of 1,600 → 10% = 160 → half = 80
Find 5% of 3,200
Step 1: 10% of 3,200 = 320
Step 2: Half of 320 = 160
Step 2: Half of 320 = 160
Answer: 160
The 15% Trick — 10% + 5%
15% = 10% + 5%
15% of 240 → 10% = 24, 5% = 12 → 24 + 12 = 36
15% of 800 → 10% = 80, 5% = 40 → 80 + 40 = 120
15% of 4,600 → 10% = 460, 5% = 230 → 460 + 230 = 690
15% of 240 → 10% = 24, 5% = 12 → 24 + 12 = 36
15% of 800 → 10% = 80, 5% = 40 → 80 + 40 = 120
15% of 4,600 → 10% = 460, 5% = 230 → 460 + 230 = 690
A shirt costs ₹1,200 and has a 15% discount. Find the discount amount.
10% of 1,200 = 120
5% of 1,200 = 60
15% = 120 + 60 = ₹180
5% of 1,200 = 60
15% = 120 + 60 = ₹180
Answer: ₹180 discount. Sale price = ₹1,020
The 25% Trick — Divide by 4
25% = ÷ 4 (one quarter)
25% of 840 → 840 ÷ 4 = 210
25% of 360 → 360 ÷ 4 = 90
25% of 1,000 → 1,000 ÷ 4 = 250
25% of 840 → 840 ÷ 4 = 210
25% of 360 → 360 ÷ 4 = 90
25% of 1,000 → 1,000 ÷ 4 = 250
The 50% Trick — Divide by 2
50% = ÷ 2 (half)
50% of 370 → 370 ÷ 2 = 185
50% of 999 → 999 ÷ 2 = 499.5
50% of 6,400 → 6,400 ÷ 2 = 3,200
50% of 370 → 370 ÷ 2 = 185
50% of 999 → 999 ÷ 2 = 499.5
50% of 6,400 → 6,400 ÷ 2 = 3,200
The 75% Trick — 50% + 25%
75% = 50% + 25% (or ¾ of the number)
75% of 840 → 50% = 420, 25% = 210 → 420 + 210 = 630
75% of 1,200 → 50% = 600, 25% = 300 → 600 + 300 = 900
75% of 840 → 50% = 420, 25% = 210 → 420 + 210 = 630
75% of 1,200 → 50% = 600, 25% = 300 → 600 + 300 = 900
The 12.5% Trick — Divide by 8
12.5% = ÷ 8 (one-eighth)
12.5% of 160 → 160 ÷ 8 = 20
12.5% of 800 → 800 ÷ 8 = 100
12.5% of 4,000 → 4,000 ÷ 8 = 500
12.5% of 160 → 160 ÷ 8 = 20
12.5% of 800 → 800 ÷ 8 = 100
12.5% of 4,000 → 4,000 ÷ 8 = 500
The 33.33% Trick — Divide by 3
33.33% = ÷ 3 (one-third)
33.33% of 900 → 900 ÷ 3 = 300
33.33% of 2,400 → 2,400 ÷ 3 = 800
66.66% = 2/3 → 66.66% of 900 = 600
33.33% of 900 → 900 ÷ 3 = 300
33.33% of 2,400 → 2,400 ÷ 3 = 800
66.66% = 2/3 → 66.66% of 900 = 600
Percentage Quick Reference Table
| Percentage | Fraction | Trick | Example (of 600) |
|---|---|---|---|
| 10% | 1/10 | Move decimal left | 60 |
| 5% | 1/20 | Half of 10% | 30 |
| 12.5% | 1/8 | Divide by 8 | 75 |
| 15% | 3/20 | 10% + 5% | 90 |
| 20% | 1/5 | Divide by 5 | 120 |
| 25% | 1/4 | Divide by 4 | 150 |
| 33.33% | 1/3 | Divide by 3 | 200 |
| 50% | 1/2 | Divide by 2 | 300 |
| 66.66% | 2/3 | 2 × (÷3) | 400 |
| 75% | 3/4 | 50% + 25% | 450 |
Reverse Percentage — Finding the Original
When a value has increased by x%, to find original: divide by (1 + x/100)
When a value has decreased by x%, to find original: divide by (1 − x/100)
When a value has decreased by x%, to find original: divide by (1 − x/100)
After a 20% increase, a price becomes ₹600. What was the original price?
Original × 1.20 = 600
Original = 600 ÷ 1.20 = ₹500
Original = 600 ÷ 1.20 = ₹500
Answer: ₹500
After a 10% decrease, a price becomes ₹450. Find the original.
Original × 0.90 = 450
Original = 450 ÷ 0.90 = ₹500
Original = 450 ÷ 0.90 = ₹500
Answer: ₹500
Combining Percentages — Successive Changes
A price increases by 20% then decreases by 10%. What is the net change?
Net effect = (1.20 × 0.90) = 1.08
Net change = +8% increase
Shortcut: a + b + (ab/100) = 20 + (−10) + (20×−10)/100 = 10 − 2 = 8%
Net change = +8% increase
Shortcut: a + b + (ab/100) = 20 + (−10) + (20×−10)/100 = 10 − 2 = 8%
Answer: 8% increase
Find 23% of 500
20% of 500 = 100
3% of 500 = 15
23% = 100 + 15 = 115
3% of 500 = 15
23% = 100 + 15 = 115
Answer: 115
Find 37.5% of 480
37.5% = 3/8
480 ÷ 8 = 60
60 × 3 = 180
480 ÷ 8 = 60
60 × 3 = 180
Answer: 180
If 35% of a number is 175, find the number.
Number = 175 ÷ 0.35 = 175 × 100/35 = 500
Answer: 500
Find 17.5% of 2,000
10% = 200, 5% = 100, 2.5% = 50
17.5% = 200 + 100 + 50 = 350
17.5% = 200 + 100 + 50 = 350
Answer: 350
A number is increased by 25% and then by 20%. What is the overall % increase?
Net = 25 + 20 + (25×20)/100 = 45 + 5 = 50%
Answer: 50% increase
What percentage of 250 is 45?
(45/250) × 100 = (45 × 2) / 5 = 90/5 = 18%
Answer: 18%
If the price of sugar increases from ₹40 to ₹50 per kg, find the percentage increase.
Increase = 50 − 40 = 10
% increase = (10/40) × 100 = 25%
% increase = (10/40) × 100 = 25%
Answer: 25%
In an exam, Rahul scored 72 out of 90. Find his percentage.
72/90 × 100 = 8/10 × 100 = 80%
Answer: 80%
Find 88% of 350
90% of 350 = 315
2% of 350 = 7
88% = 315 − 7 = 308
(Trick: 88% = 90% − 2%)
2% of 350 = 7
88% = 315 − 7 = 308
(Trick: 88% = 90% − 2%)
Answer: 308
Two successive discounts of 20% and 10% are equivalent to what single discount?
Effective = 20 + 10 − (20×10)/100 = 30 − 2 = 28%
(For discounts, we subtract ab/100)
(For discounts, we subtract ab/100)
Answer: 28% single discount
🎯 SSC-Level Competitive Problems
SSC 1: If A's income is 25% more than B's income, then B's income is how much percent less than A's?
If B = 100, then A = 125
Difference = 25
% less = (25/125) × 100 = 20%
Shortcut: x% more → x/(100+x) × 100 % less = 25/125 × 100 = 20%
Difference = 25
% less = (25/125) × 100 = 20%
Shortcut: x% more → x/(100+x) × 100 % less = 25/125 × 100 = 20%
Answer: 20% less
SSC 2: The population of a town increases by 10% every year. If the current population is 24,200, what was it 2 years ago?
P × (1.10)² = 24,200
P × 1.21 = 24,200
P = 24,200 ÷ 1.21 = 20,000
P × 1.21 = 24,200
P = 24,200 ÷ 1.21 = 20,000
Answer: 20,000
SSC 3: In an election, candidate A gets 60% of valid votes. 15% of total votes are invalid. A gets 5,100 votes. Find total votes.
Valid votes = 85% of total
A's votes = 60% of 85% of total = 0.60 × 0.85 × T = 0.51T
0.51T = 5,100 → T = 10,000
A's votes = 60% of 85% of total = 0.60 × 0.85 × T = 0.51T
0.51T = 5,100 → T = 10,000
Answer: 10,000 total votes
SSC 4: A shopkeeper marks his goods 40% above cost price and gives 20% discount. Find his profit %.
Let CP = 100. MP = 140
SP = 140 × 0.80 = 112
Profit = 12, Profit% = 12%
SP = 140 × 0.80 = 112
Profit = 12, Profit% = 12%
Answer: 12% profit
SSC 5: A man spends 75% of his income. If his income increases by 20% and expenses increase by 10%, find % change in savings.
Let income = 100, expenses = 75, savings = 25
New income = 120, new expenses = 82.5, new savings = 37.5
Change in savings = (37.5 − 25)/25 × 100 = 50% increase
New income = 120, new expenses = 82.5, new savings = 37.5
Change in savings = (37.5 − 25)/25 × 100 = 50% increase
Answer: 50% increase in savings
Practice Problems — Percentages
P9.1
Find 15% of 680
10% of 680 = 68, 5% = 34. So 15% = 68 + 34 = 102
P9.2
Find 25% of 1,360
1,360 ÷ 4 = 340
P9.3
Find 12.5% of 720
720 ÷ 8 = 90
P9.4
Find 33.33% of 1,800
1,800 ÷ 3 = 600
P9.5
Find 75% of 960
50% = 480, 25% = 240. 75% = 480 + 240 = 720
P9.6
A price increases from ₹80 to ₹100. What is the percentage increase?
Increase = 20. % = (20/80) × 100 = 25%
P9.7
After a 30% discount, a bag costs ₹560. Find the original price.
Original × 0.70 = 560. Original = 560 ÷ 0.70 = ₹800
P9.8
Find 47% of 200
50% = 100, 3% = 6. 47% = 100 − 6 = 94
P9.9
Successive discounts of 10% and 20% equal what single discount?
10 + 20 − (10×20)/100 = 30 − 2 = 28%
P9.10
A's salary is 20% less than B's. B's salary is what % more than A's?
If B=100, A=80. Diff=20. 20/80 × 100 = 25%
P9.11
Find 62.5% of 480
62.5% = 5/8. 480 ÷ 8 = 60. 60 × 5 = 300
P9.12
If 40% of a number is 260, find the number.
260 ÷ 0.40 = 650
P9.13
Find 87.5% of 640
87.5% = 7/8. 640 ÷ 8 = 80. 80 × 7 = 560
P9.14
A number is increased by 10% and then decreased by 10%. What is the net change?
Net = 10 − 10 − (10×10)/100 = −1. 1% decrease
P9.15
In a class of 60 students, 45 passed. What percentage failed?
Failed = 15. 15/60 × 100 = 25%
Chapter 10
🥧 Fractions & Decimals — Vedic Speed Methods
🔥 Fractions Made Lightning Fast
Most students waste precious minutes on fraction arithmetic. With these Vedic techniques, you'll compare, add, and convert fractions in seconds!
Fraction → Decimal Reference Table
Memorize these conversions. They appear everywhere in competitive exams!
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| 1/2 | 0.5 | 1/11 | 0.0909... |
| 1/3 | 0.333... | 1/12 | 0.0833... |
| 1/4 | 0.25 | 1/13 | 0.0769... |
| 1/5 | 0.2 | 1/14 | 0.0714... |
| 1/6 | 0.1666... | 1/15 | 0.0666... |
| 1/7 | 0.142857... | 1/16 | 0.0625 |
| 1/8 | 0.125 | 1/17 | 0.0588... |
| 1/9 | 0.111... | 1/18 | 0.0555... |
| 1/10 | 0.1 | 1/19 | 0.0526... |
| — | — | 1/20 | 0.05 |
The decimal expansion of 1/7 = 0.142857142857... is called a cyclic number. Multiply 142857 by 2, 3, 4, 5, or 6 — you get the SAME digits in a different order! 142857 × 2 = 285714, × 3 = 428571, and so on. Amazing!
Comparing Fractions — Cross-Multiply Method
To compare a/b and c/d, cross-multiply: compare a×d with c×b.
Cross-Multiply to Compare:
Compare 3/7 vs 5/11:
3 × 11 = 33 vs 5 × 7 = 35
Since 33 < 35 → 3/7 < 5/11 → 5/11 is bigger!
Compare 3/7 vs 5/11:
3 × 11 = 33 vs 5 × 7 = 35
Since 33 < 35 → 3/7 < 5/11 → 5/11 is bigger!
Which is larger: 5/8 or 7/11?
5 × 11 = 55 vs 7 × 8 = 56
55 < 56
55 < 56
7/11 is larger
Which is larger: 4/9 or 3/7?
4 × 7 = 28 vs 3 × 9 = 27
28 > 27
28 > 27
4/9 is larger
Adding Fractions — Butterfly Method
For a/b + c/d, use: (a×d + c×b) / (b×d)
Butterfly Method:
2/3 + 3/5 = (2×5 + 3×3) / (3×5) = (10 + 9) / 15 = 19/15
Visually: cross-multiply for the numerator, multiply across for denominator!
2/3 + 3/5 = (2×5 + 3×3) / (3×5) = (10 + 9) / 15 = 19/15
Visually: cross-multiply for the numerator, multiply across for denominator!
Find 3/4 + 2/7
(3×7 + 2×4) / (4×7) = (21 + 8) / 28 = 29/28
Answer: 29/28 = 1 1/28
Find 5/6 − 1/4
(5×4 − 1×6) / (6×4) = (20 − 6) / 24 = 14/24 = 7/12
Answer: 7/12
Multiplying Fractions — Cancel First!
Before multiplying, cancel common factors diagonally:
4/9 × 3/8
Cancel: 4 and 8 (÷4) → 1 and 2; 3 and 9 (÷3) → 1 and 3
Result: 1/3 × 1/2 = 1/6
4/9 × 3/8
Cancel: 4 and 8 (÷4) → 1 and 2; 3 and 9 (÷3) → 1 and 3
Result: 1/3 × 1/2 = 1/6
Find 7/12 × 8/21
Cancel: 7 with 21 (÷7) → 1 and 3; 8 with 12 (÷4) → 2 and 3
Result: 1/3 × 2/3 = 2/9
Result: 1/3 × 2/3 = 2/9
Answer: 2/9
Mixed Number Operations
Find 3 1/2 + 2 3/4
Whole parts: 3 + 2 = 5
Fractions: 1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4
Total: 5 + 1 1/4 = 6 1/4
Fractions: 1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4
Total: 5 + 1 1/4 = 6 1/4
Answer: 6 1/4 = 6.25
Find 5 2/3 − 2 5/6
Convert: 17/3 − 17/6 = 34/6 − 17/6 = 17/6 = 2 5/6
Answer: 2 5/6
Fraction Visualizations
1/4 = 25%
1/3 = 33.3%
1/2 = 50%
5/8 = 62.5%
3/4 = 75%
Convert 0.375 to a fraction
0.375 = 375/1000 = 3/8
Answer: 3/8
Arrange in ascending order: 3/5, 2/3, 5/8
Convert to decimals: 3/5 = 0.6, 2/3 = 0.667, 5/8 = 0.625
Order: 0.6 < 0.625 < 0.667
Order: 0.6 < 0.625 < 0.667
Answer: 3/5 < 5/8 < 2/3
Find 2/5 of 3/7 of 490
3/7 of 490 = 210
2/5 of 210 = 84
2/5 of 210 = 84
Answer: 84
Simplify: (5/6 + 1/3) ÷ (7/4 − 1/2)
5/6 + 1/3 = 5/6 + 2/6 = 7/6
7/4 − 1/2 = 7/4 − 2/4 = 5/4
(7/6) ÷ (5/4) = 7/6 × 4/5 = 28/30 = 14/15
7/4 − 1/2 = 7/4 − 2/4 = 5/4
(7/6) ÷ (5/4) = 7/6 × 4/5 = 28/30 = 14/15
Answer: 14/15
Practice Problems — Fractions & Decimals
P10.1
Which is larger: 5/9 or 4/7?
5×7=35, 4×9=36. 35 < 36, so 4/7 is larger
P10.2
Find 2/5 + 3/8
(2×8 + 3×5)/(5×8) = (16+15)/40 = 31/40
P10.3
Find 7/8 − 2/3
(7×3 − 2×8)/(8×3) = (21−16)/24 = 5/24
P10.4
Convert 0.625 to a fraction
625/1000 = 5/8
P10.5
Find 5/6 × 12/25
Cancel: 5↔25(÷5)=1,5; 12↔6(÷6)=2,1. 1/1 × 2/5 = 2/5
P10.6
Find 4 1/3 + 2 5/6
13/3 + 17/6 = 26/6 + 17/6 = 43/6 = 7 1/6
P10.7
Which is larger: 7/13 or 8/15?
7×15=105, 8×13=104. 105 > 104, so 7/13 is larger
P10.8
Find 3/4 of 2/3 of 120
2/3 of 120 = 80. 3/4 of 80 = 60
P10.9
Convert 2.875 to a fraction
2.875 = 2 + 875/1000 = 2 + 7/8 = 23/8
P10.10
Simplify: 3/4 ÷ 9/16
3/4 × 16/9 = (3×16)/(4×9) = 48/36 = 4/3 = 1 1/3
Chapter 11
🔲 Squares & Square Roots — Vedic Shortcuts
🔥 Square Any Number in Seconds
Forget memorizing tables endlessly. Vedic Mathematics gives you magical methods to find the square of ANY number — whether it's near 50, near 100, or any random number. These tricks will blow your mind!
Perfect Squares 1–30 Reference
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 11 | 121 | 21 | 441 |
| 2 | 4 | 12 | 144 | 22 | 484 |
| 3 | 9 | 13 | 169 | 23 | 529 |
| 4 | 16 | 14 | 196 | 24 | 576 |
| 5 | 25 | 15 | 225 | 25 | 625 |
| 6 | 36 | 16 | 256 | 26 | 676 |
| 7 | 49 | 17 | 289 | 27 | 729 |
| 8 | 64 | 18 | 324 | 28 | 784 |
| 9 | 81 | 19 | 361 | 29 | 841 |
| 10 | 100 | 20 | 400 | 30 | 900 |
Notice the pattern of last digits in perfect squares: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 — and it repeats! A perfect square can ONLY end in 0, 1, 4, 5, 6, or 9. It can NEVER end in 2, 3, 7, or 8!
Squaring Numbers Near 50 — The (50±a)² Method
Formula: (50 + a)² = (25 + a) × 100 + a²
The left part: 25 + a (or 25 − a for numbers below 50)
The right part: a² (always write as 2 digits with leading zero if needed)
The left part: 25 + a (or 25 − a for numbers below 50)
The right part: a² (always write as 2 digits with leading zero if needed)
Find 48²
48 = 50 − 2, so a = −2
Left part: 25 + (−2) = 23
Right part: (−2)² = 04
Combine: 2304
Left part: 25 + (−2) = 23
Right part: (−2)² = 04
Combine: 2304
48² = 2,304 ✓
Find 53²
53 = 50 + 3, so a = 3
Left part: 25 + 3 = 28
Right part: 3² = 09
Combine: 2809
Left part: 25 + 3 = 28
Right part: 3² = 09
Combine: 2809
53² = 2,809 ✓
Find 46²
46 = 50 − 4, so a = −4
Left: 25 − 4 = 21
Right: 4² = 16
Combine: 2116
Left: 25 − 4 = 21
Right: 4² = 16
Combine: 2116
46² = 2,116 ✓
Find 57²
57 = 50 + 7, so a = 7
Left: 25 + 7 = 32
Right: 7² = 49
Combine: 3249
Left: 25 + 7 = 32
Right: 7² = 49
Combine: 3249
57² = 3,249 ✓
When a² gives a number larger than 99 (like a=8, a²=64... wait that's fine). But for a=10: a²=100. In that case, write 00 and carry 1 to the left part. Example: 60² → Left: 25+10=35, right: 100→00 carry 1 → 36|00 = 3600. ✓
Squaring Numbers Near 100 — Nikhilam Method
For numbers near 100:
n² = [n + (n − 100)] × 100 + (n − 100)²
Or simpler: Find deficit/surplus from 100. Add it to the number (left part). Square the deficit/surplus (right part, 2 digits).
n² = [n + (n − 100)] × 100 + (n − 100)²
Or simpler: Find deficit/surplus from 100. Add it to the number (left part). Square the deficit/surplus (right part, 2 digits).
Find 96²
Deficit = 100 − 96 = 4
Left: 96 − 4 = 92
Right: 4² = 16
Combine: 9216
Left: 96 − 4 = 92
Right: 4² = 16
Combine: 9216
96² = 9,216 ✓
Find 104²
Surplus = 104 − 100 = 4
Left: 104 + 4 = 108
Right: 4² = 16
Combine: 10816
Left: 104 + 4 = 108
Right: 4² = 16
Combine: 10816
104² = 10,816 ✓
Find 93²
Deficit = 7
Left: 93 − 7 = 86
Right: 7² = 49
Combine: 8649
Left: 93 − 7 = 86
Right: 7² = 49
Combine: 8649
93² = 8,649 ✓
Find 112²
Surplus = 12
Left: 112 + 12 = 124
Right: 12² = 144 → write 44, carry 1
Left becomes: 124 + 1 = 125
Combine: 12544
Left: 112 + 12 = 124
Right: 12² = 144 → write 44, carry 1
Left becomes: 124 + 1 = 125
Combine: 12544
112² = 12,544 ✓
Duplex Method (Dwandwa Yoga) — Square Any Number
Duplex (D) of digits:
• Single digit a: D(a) = a²
• Two digits ab: D(ab) = 2 × a × b
• Three digits abc: D(abc) = 2ac + b²
For squaring, compute duplexes from right, carry as needed.
• Single digit a: D(a) = a²
• Two digits ab: D(ab) = 2 × a × b
• Three digits abc: D(abc) = 2ac + b²
For squaring, compute duplexes from right, carry as needed.
Find 34² using Duplex
Digits: 3, 4
Units: D(4) = 16 → write 6, carry 1
Tens: D(3,4) = 2×3×4 = 24 + 1 = 25 → write 5, carry 2
Hundreds: D(3) = 9 + 2 = 11
Result: 1156
Units: D(4) = 16 → write 6, carry 1
Tens: D(3,4) = 2×3×4 = 24 + 1 = 25 → write 5, carry 2
Hundreds: D(3) = 9 + 2 = 11
Result: 1156
34² = 1,156 ✓
Find 123² using Duplex
Digits: 1, 2, 3
Units: D(3) = 9
Tens: D(2,3) = 12 → write 2, carry 1
Hundreds: D(1,2,3) = 2×1×3 + 2² = 6+4 = 10 + 1 = 11 → write 1, carry 1
Thousands: D(1,2) = 2×1×2 = 4 + 1 = 5
Ten-thousands: D(1) = 1
Result: 15129
Units: D(3) = 9
Tens: D(2,3) = 12 → write 2, carry 1
Hundreds: D(1,2,3) = 2×1×3 + 2² = 6+4 = 10 + 1 = 11 → write 1, carry 1
Thousands: D(1,2) = 2×1×2 = 4 + 1 = 5
Ten-thousands: D(1) = 1
Result: 15129
123² = 15,129 ✓
Square Roots by Observation
Last digit of perfect square → possible last digits of root:
Ends in 1 → root ends in 1 or 9
Ends in 4 → root ends in 2 or 8
Ends in 5 → root ends in 5
Ends in 6 → root ends in 4 or 6
Ends in 9 → root ends in 3 or 7
Ends in 0 → root ends in 0
Ends in 1 → root ends in 1 or 9
Ends in 4 → root ends in 2 or 8
Ends in 5 → root ends in 5
Ends in 6 → root ends in 4 or 6
Ends in 9 → root ends in 3 or 7
Ends in 0 → root ends in 0
Find √2025
Ends in 5 → root ends in 5
2025 is between 40²=1600 and 50²=2500
So root is between 40 and 50, ending in 5 → root = 45
2025 is between 40²=1600 and 50²=2500
So root is between 40 and 50, ending in 5 → root = 45
√2025 = 45 ✓
Find √5476
Ends in 6 → root ends in 4 or 6
5476 is between 70²=4900 and 80²=6400
Try 74: 74² = 5476 ✓ (using near-100 method: 74−26=48→left, 26²=676→right... or just check: 74²=(75−1)²=5625−150+1=5476 ✓)
5476 is between 70²=4900 and 80²=6400
Try 74: 74² = 5476 ✓ (using near-100 method: 74−26=48→left, 26²=676→right... or just check: 74²=(75−1)²=5625−150+1=5476 ✓)
√5476 = 74 ✓
Find √7921
Ends in 1 → root ends in 1 or 9
7921 is between 80²=6400 and 90²=8100
Try 89: 89² = (90−1)² = 8100−180+1 = 7921 ✓
7921 is between 80²=6400 and 90²=8100
Try 89: 89² = (90−1)² = 8100−180+1 = 7921 ✓
√7921 = 89 ✓
Quick check: If the number before the last two digits is closer to the lower perfect square, pick the lower option. If closer to the higher, pick the higher one. For √5476: 54 is closer to 49 (7²) than 64 (8²), so try 74 first rather than 76.
Practice Problems — Squares & Square Roots
P11.1
Find 47² (near 50 method)
25−3=22, 3²=09. 2209
P11.2
Find 54² (near 50 method)
25+4=29, 4²=16. 2916
P11.3
Find 97² (near 100 method)
97−3=94, 3²=09. 9409
P11.4
Find 106² (near 100 method)
106+6=112, 6²=36. 11236
P11.5
Find √3969
Ends in 9→root ends in 3 or 7. Between 60²=3600 and 70²=4900. Try 63: 63²=3969 ✓. 63
P11.6
Find 42² (near 50 method)
25−8=17, 8²=64. 1764
P11.7
Find 58² (near 50 method)
25+8=33, 8²=64. 3364
P11.8
Find 91² (near 100 method)
91−9=82, 9²=81. 8281
P11.9
Find √6561
Ends in 1→root ends in 1 or 9. Between 80² and 90². Try 81: 81²=6561 ✓. 81
P11.10
Find 45² (mentally!)
Numbers ending in 5: n5² = n×(n+1) | 25. 4×5=20, append 25. 2025
P11.11
Find 65²
6×7=42, append 25. 4225
P11.12
Find 108²
108+8=116, 8²=64. 11664
P11.13
Find √4624 (not a perfect square — find nearest)
67²=4489, 68²=4624. √4624 = 68
P11.14
Find 39² using Duplex method
D(9)=81→write 1 carry 8. D(3,9)=54+8=62→write 2 carry 6. D(3)=9+6=15. 1521
P11.15
Find √1849
Ends in 9→root ends in 3 or 7. Between 40² and 50². Try 43: 43²=1849 ✓. 43