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
Chapter 12
🧊 Cubes & Cube Roots — Vedic Mastery
🔥 Cube Roots in 5 Seconds
While others struggle with long calculations, you'll extract cube roots of perfect cubes by just looking at them. The Vedic method uses a simple last-digit mapping and range estimation. Let's unlock this superpower!
Perfect Cubes 1–20 Reference
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 11 | 1,331 |
| 2 | 8 | 12 | 1,728 |
| 3 | 27 | 13 | 2,197 |
| 4 | 64 | 14 | 2,744 |
| 5 | 125 | 15 | 3,375 |
| 6 | 216 | 16 | 4,096 |
| 7 | 343 | 17 | 4,913 |
| 8 | 512 | 18 | 5,832 |
| 9 | 729 | 19 | 6,859 |
| 10 | 1,000 | 20 | 8,000 |
Cube Root by Observation — Last Digit Mapping
Last digit of cube → Last digit of cube root:
Pattern: 1,4,5,6,9,0 stay the same. 2↔8 and 3↔7 are complements of 10!
| Cube ends in | Root ends in | Memory Trick |
|---|---|---|
| 1 | 1 | Same |
| 2 | 8 | Complement to 10 |
| 3 | 7 | Complement to 10 |
| 4 | 4 | Same |
| 5 | 5 | Same |
| 6 | 6 | Same |
| 7 | 3 | Complement to 10 |
| 8 | 2 | Complement to 10 |
| 9 | 9 | Same |
| 0 | 0 | Same |
Pattern: 1,4,5,6,9,0 stay the same. 2↔8 and 3↔7 are complements of 10!
Finding Cube Roots — Step-by-Step
Step 1: Look at the last digit → determine last digit of root (using mapping above)
Step 2: Remove the last 3 digits. The remaining number falls between two consecutive cubes → that gives you the tens digit of the root.
Step 2: Remove the last 3 digits. The remaining number falls between two consecutive cubes → that gives you the tens digit of the root.
Find ∛17576
Step 1: Last digit = 6 → root ends in 6
Step 2: Remove last 3 digits: 17 remains
2³ = 8, 3³ = 27. Since 8 < 17 < 27, tens digit = 2
Cube root = 26
Step 2: Remove last 3 digits: 17 remains
2³ = 8, 3³ = 27. Since 8 < 17 < 27, tens digit = 2
Cube root = 26
∛17576 = 26 ✓
Find ∛42875
Step 1: Last digit = 5 → root ends in 5
Step 2: Remove last 3 digits: 42 remains
3³ = 27, 4³ = 64. Since 27 < 42 < 64, tens digit = 3
Cube root = 35
Step 2: Remove last 3 digits: 42 remains
3³ = 27, 4³ = 64. Since 27 < 42 < 64, tens digit = 3
Cube root = 35
∛42875 = 35 ✓
Find ∛103823
Step 1: Last digit = 3 → root ends in 7
Step 2: Remove last 3 digits: 103 remains
4³ = 64, 5³ = 125. Since 64 < 103 < 125, tens digit = 4
Cube root = 47
Step 2: Remove last 3 digits: 103 remains
4³ = 64, 5³ = 125. Since 64 < 103 < 125, tens digit = 4
Cube root = 47
∛103823 = 47 ✓
Find ∛274625
Step 1: Last digit = 5 → root ends in 5
Step 2: Remove last 3 digits: 274 remains
6³ = 216, 7³ = 343. Since 216 < 274 < 343, tens digit = 6
Cube root = 65
Step 2: Remove last 3 digits: 274 remains
6³ = 216, 7³ = 343. Since 216 < 274 < 343, tens digit = 6
Cube root = 65
∛274625 = 65 ✓
Anurupya Sutra for Cubing — The Ratio Method
To cube a 2-digit number (10a + b):
1. Write four terms in geometric ratio: a³, a²b, ab², b³
2. Double the two middle terms (add them to themselves)
3. Add with carry from right to left
1. Write four terms in geometric ratio: a³, a²b, ab², b³
2. Double the two middle terms (add them to themselves)
3. Add with carry from right to left
Find 23³ using Anurupya
Step 1 — Four terms: 2³=8, 2²×3=12, 2×3²=18, 3³=27
Step 2 — Double middle: 8, 12+12=24, 18+18=36, 27
So: 8, 24, 36, 27
Step 3 — Carry:
Units: 27 → write 7, carry 2
Tens: 36+2=38 → write 8, carry 3
Hundreds: 24+3=27 → write 7, carry 2
Thousands: 8+2=10 → write 10
Remaining: 10
Result: 12167
Step 2 — Double middle: 8, 12+12=24, 18+18=36, 27
So: 8, 24, 36, 27
Step 3 — Carry:
Units: 27 → write 7, carry 2
Tens: 36+2=38 → write 8, carry 3
Hundreds: 24+3=27 → write 7, carry 2
Thousands: 8+2=10 → write 10
Remaining: 10
Result: 12167
23³ = 12,167 ✓
Find 14³ using Anurupya
Four terms: 1, 4, 16, 64
Double middle: 1, 8, 32, 64
Carry:
64→4, carry 6
32+6=38→8, carry 3
8+3=11→1, carry 1
1+1=2
Result: 2744
Double middle: 1, 8, 32, 64
Carry:
64→4, carry 6
32+6=38→8, carry 3
8+3=11→1, carry 1
1+1=2
Result: 2744
14³ = 2,744 ✓
Find 31³ using Anurupya
Four terms: 27, 9, 3, 1
Double middle: 27, 18, 6, 1
Carry:
1→1
6→6
18→8, carry 1
27+1=28
Result: 28|8|6|1 = 29791
Double middle: 27, 18, 6, 1
Carry:
1→1
6→6
18→8, carry 1
27+1=28
Result: 28|8|6|1 = 29791
31³ = 29,791 ✓
Find 15³ using Anurupya
Four terms: 1, 5, 25, 125
Double middle: 1, 10, 50, 125
Carry:
125→5 carry 12
50+12=62→2 carry 6
10+6=16→6 carry 1
1+1=2
No wait, let me redo: 1, 10, 50, 125
Units: 125 → 5, carry 12
Tens: 50+12 = 62 → 2, carry 6
Hundreds: 10+6 = 16 → 6, carry 1
Thousands: 1+1 = 2
..but wait, let me re-examine: 3|3|7|5. Actually: 3375
Double middle: 1, 10, 50, 125
Carry:
125→5 carry 12
50+12=62→2 carry 6
10+6=16→6 carry 1
1+1=2
No wait, let me redo: 1, 10, 50, 125
Units: 125 → 5, carry 12
Tens: 50+12 = 62 → 2, carry 6
Hundreds: 10+6 = 16 → 6, carry 1
Thousands: 1+1 = 2
..but wait, let me re-examine: 3|3|7|5. Actually: 3375
15³ = 3,375 ✓
Practice Problems — Cubes & Cube Roots
P12.1
Find ∛9261
Last digit 1→root ends in 1. Remove 261: 9 remains. 2³=8 < 9 < 27=3³. Tens=2. Root=21
P12.2
Find ∛32768
Last digit 8→root ends in 2. Remove 768: 32. 3³=27 < 32 < 64=4³. Tens=3. Root=32
P12.3
Find ∛68921
Last digit 1→root ends in 1. Remove 921: 68. 4³=64 < 68 < 125=5³. Tens=4. Root=41
P12.4
Find 12³ using Anurupya
Terms: 1, 2, 4, 8. Double: 1, 4, 8, 8. Carry: 8→8, 8→8, 4→4, 1→1. 1728
P12.5
Find ∛185193
Last digit 3→root ends in 7. Remove 193: 185. 5³=125 < 185 < 216=6³. Tens=5. Root=57
P12.6
Find 25³ using Anurupya
Terms: 8, 20, 50, 125. Double: 8, 40, 100, 125. Carry gives 15625
P12.7
Find ∛512000
512000 = 512 × 1000. ∛512=8, ∛1000=10. Root = 8×10 = 80
P12.8
Find ∛13824
Last digit 4→root ends in 4. Remove 824: 13. 2³=8 < 13 < 27=3³. Tens=2. Root=24
P12.9
Find 16³
Terms: 1, 6, 36, 216. Double: 1, 12, 72, 216. Carry gives 4096
P12.10
Find ∛250047
Last digit 7→root ends in 3. Remove 047: 250. 6³=216 < 250 < 343=7³. Tens=6. Root=63
Chapter 13
📐 Algebra — The Vedic Way
🔥 Algebra Made Visual & Instant
Algebraic identities aren't just formulas — they're visual patterns that you can SEE. The Vedic approach makes factoring, expanding, and simplifying feel like second nature!
(a + b)² = a² + 2ab + b² — Visual Square Method
Imagine a square with side (a + b). It can be divided into 4 parts:
a²
a × b
a × b
b²
Total Area = a² + ab + ab + b² = a² + 2ab + b²
Find 52² using (a+b)²
52 = 50 + 2, so a=50, b=2
a² = 2500, 2ab = 200, b² = 4
52² = 2500 + 200 + 4 = 2704
a² = 2500, 2ab = 200, b² = 4
52² = 2500 + 200 + 4 = 2704
Answer: 2,704
Find 103²
103 = 100 + 3
100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609
100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609
Answer: 10,609
(a − b)² = a² − 2ab + b²
Find 48² using (a−b)²
48 = 50 − 2, so a=50, b=2
50² − 2(50)(2) + 2² = 2500 − 200 + 4 = 2304
50² − 2(50)(2) + 2² = 2500 − 200 + 4 = 2304
Answer: 2,304
Find 97²
97 = 100 − 3
100² − 2(100)(3) + 3² = 10000 − 600 + 9 = 9409
100² − 2(100)(3) + 3² = 10000 − 600 + 9 = 9409
Answer: 9,409
a² − b² = (a+b)(a−b) — Instant Factoring
The Difference of Squares identity is a SUPERPOWER!
Any time you see x² − y², instantly factor it as (x+y)(x−y).
This turns hard calculations into easy mental math!
Any time you see x² − y², instantly factor it as (x+y)(x−y).
This turns hard calculations into easy mental math!
Find 67² − 33²
(67+33)(67−33) = 100 × 34 = 3400
Done in 3 seconds!
Done in 3 seconds!
Answer: 3,400
Find 85² − 15²
(85+15)(85−15) = 100 × 70 = 7000
Answer: 7,000
Find 123² − 77²
(123+77)(123−77) = 200 × 46 = 9200
Answer: 9,200
Find 1.7² − 0.3²
(1.7+0.3)(1.7−0.3) = 2.0 × 1.4 = 2.8
Answer: 2.8
Vertically & Crosswise (Urdhva Tiryak) — Polynomial Multiplication
Multiply (ax + b)(cx + d) Vedic way:
• x² coefficient: a × c (vertical)
• x coefficient: ad + bc (crosswise)
• constant: b × d (vertical)
• x² coefficient: a × c (vertical)
• x coefficient: ad + bc (crosswise)
• constant: b × d (vertical)
Expand (2x + 3)(4x + 5)
x² term: 2×4 = 8x²
x term: 2×5 + 3×4 = 10+12 = 22x
constant: 3×5 = 15
Result: 8x² + 22x + 15
x term: 2×5 + 3×4 = 10+12 = 22x
constant: 3×5 = 15
Result: 8x² + 22x + 15
Answer: 8x² + 22x + 15
Expand (3x − 2)(x + 7)
x²: 3×1 = 3x²
x: 3×7 + (−2)×1 = 21−2 = 19x
constant: (−2)×7 = −14
Result: 3x² + 19x − 14
x: 3×7 + (−2)×1 = 21−2 = 19x
constant: (−2)×7 = −14
Result: 3x² + 19x − 14
Answer: 3x² + 19x − 14
Paravartya Sutra — Division of Polynomials
Paravartya Yojayet = "Transpose and Adjust"
For dividing by (x − a), use 'a' as a seed and perform synthetic division!
For dividing by (x − a), use 'a' as a seed and perform synthetic division!
Divide (x³ + 3x² − 4x + 2) by (x − 1)
Seed = 1 (transpose sign of −1)
Coefficients: 1, 3, −4, 2
Bring down 1
1×1=1, add to 3→4
1×4=4, add to −4→0
1×0=0, add to 2→2
Quotient: x² + 4x + 0, Remainder: 2
Coefficients: 1, 3, −4, 2
Bring down 1
1×1=1, add to 3→4
1×4=4, add to −4→0
1×0=0, add to 2→2
Quotient: x² + 4x + 0, Remainder: 2
Answer: x² + 4x with remainder 2
Factorization Tricks
Factorize x² + 5x + 6
Find two numbers that multiply to 6 and add to 5: 2 and 3
= (x + 2)(x + 3)
= (x + 2)(x + 3)
Answer: (x + 2)(x + 3)
Factorize x² − 7x + 12
Multiply to 12, add to −7: −3 and −4
= (x − 3)(x − 4)
= (x − 3)(x − 4)
Answer: (x − 3)(x − 4)
Factorize 2x² + 7x + 3
Product = 2×3=6. Sum=7. Numbers: 6 and 1
2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)
2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)
Answer: (2x + 1)(x + 3)
Practice Problems — Algebra
P13.1
Find 72² − 28² using identity
(72+28)(72−28) = 100 × 44 = 4400
P13.2
Expand (5x + 2)(3x − 4)
15x² − 20x + 6x − 8 = 15x² − 14x − 8
P13.3
Find 61² using (60+1)²
3600 + 120 + 1 = 3721
P13.4
Factorize x² − 9x + 20
Numbers: −4, −5 (product=20, sum=−9). (x−4)(x−5)
P13.5
Find 99² using (100−1)²
10000 − 200 + 1 = 9801
P13.6
Find 53² − 47²
(53+47)(53−47) = 100 × 6 = 600
P13.7
Factorize x² + 11x + 30
5 and 6 (product=30, sum=11). (x+5)(x+6)
P13.8
Expand (x + 3)³
x³ + 3(x²)(3) + 3(x)(9) + 27 = x³ + 9x² + 27x + 27
P13.9
Find 205² − 195²
(205+195)(205−195) = 400 × 10 = 4000
P13.10
Factorize 3x² + 10x + 8
Product=24, sum=10. Numbers: 4,6. 3x²+6x+4x+8 = 3x(x+2)+4(x+2) = (3x+4)(x+2)
Chapter 14
🏆 Speed Mathematics for Competitive Exams
🔥 Exam Hall Weapons
This chapter is your arsenal for SSC, Banking, JEE, and Olympiad exams. Every technique here saves precious minutes in the exam hall. Master these, and you'll finish the quantitative section with time to spare!
Simplification Shortcuts — BODMAS with Vedic Speed
Simplify: 36 × 25 + 64 × 25
Take 25 common: 25 × (36 + 64) = 25 × 100 = 2500
Answer: 2,500
Simplify: 847 × 847 + 153 × 153 + 2 × 847 × 153
This is a² + b² + 2ab = (a + b)²
= (847 + 153)² = 1000² = 1,000,000
= (847 + 153)² = 1000² = 1,000,000
Answer: 1,000,000
Approximation Techniques — Round & Adjust
Near-value multiplication:
a × b where a ≈ b ≈ n: use (n + d₁)(n + d₂) = n² + n(d₁+d₂) + d₁d₂
Often d₁d₂ is negligible!
a × b where a ≈ b ≈ n: use (n + d₁)(n + d₂) = n² + n(d₁+d₂) + d₁d₂
Often d₁d₂ is negligible!
Approximate 3.97 × 4.03
= (4 − 0.03)(4 + 0.03) = 4² − 0.03² = 16 − 0.0009 ≈ 15.9991 ≈ 16
Answer: ≈ 15.9991
Approximate 49 × 51
= (50−1)(50+1) = 2500 − 1 = 2499
Answer: 2,499
Approximate 998 × 1002
= (1000−2)(1000+2) = 1,000,000 − 4 = 999,996
Answer: 999,996
Ratio & Proportion Shortcuts
If A:B = 3:4 and B:C = 5:6, find A:B:C
Make B equal: A:B = 15:20, B:C = 20:24
A:B:C = 15:20:24
A:B:C = 15:20:24
Answer: 15:20:24
Divide ₹1,050 among A, B, C in ratio 2:3:5
Total parts = 10
A = 2/10 × 1050 = 210
B = 3/10 × 1050 = 315
C = 5/10 × 1050 = 525
A = 2/10 × 1050 = 210
B = 3/10 × 1050 = 315
C = 5/10 × 1050 = 525
Answer: ₹210, ₹315, ₹525
Profit & Loss — Fraction Equivalents
Use fraction equivalents for instant calculation:
10% = 1/10 | 20% = 1/5 | 25% = 1/4 | 33⅓% = 1/3
SP = CP × (1 + Profit%) | SP = CP × (1 − Loss%)
If goods are sold at x% profit: SP/CP = (100+x)/100
10% = 1/10 | 20% = 1/5 | 25% = 1/4 | 33⅓% = 1/3
SP = CP × (1 + Profit%) | SP = CP × (1 − Loss%)
If goods are sold at x% profit: SP/CP = (100+x)/100
SSC: A man buys an article for ₹800 and sells at 25% profit. Find SP.
25% profit = ¼ of CP extra
Profit = 800/4 = 200
SP = 800 + 200 = ₹1,000
Profit = 800/4 = 200
SP = 800 + 200 = ₹1,000
Answer: ₹1,000
SSC: If CP = ₹450 and SP = ₹540, find profit %.
Profit = 90. Profit% = 90/450 × 100 = 20%
Answer: 20%
SSC: By selling at ₹570, a man gains 14.28%. Find CP.
14.28% ≈ 1/7. So SP = 8/7 × CP
CP = 570 × 7/8 = ₹498.75
Or: CP = 570/1.1428 ≈ ₹499
CP = 570 × 7/8 = ₹498.75
Or: CP = 570/1.1428 ≈ ₹499
Answer: ≈ ₹499
SSC: A sells to B at 20% profit, B sells to C at 25% profit. If C pays ₹900, what did A pay?
C's price = A's price × 1.2 × 1.25 = A × 1.5
A = 900/1.5 = ₹600
A = 900/1.5 = ₹600
Answer: ₹600
SSC: If selling price is doubled, profit triples. Find profit %.
Let CP=C, SP=S, P=S−C
New: 2S−C = 3(S−C) → 2S−C = 3S−3C → 2C=S
Profit = S−C = C. Profit% = C/C × 100 = 100%
New: 2S−C = 3(S−C) → 2S−C = 3S−3C → 2C=S
Profit = S−C = C. Profit% = C/C × 100 = 100%
Answer: 100%
Time & Work — LCM Method
LCM Method: Take LCM of days as total work units.
Each person's rate = total work / their days.
Combined rate = sum of individual rates.
Each person's rate = total work / their days.
Combined rate = sum of individual rates.
Banking: A can do a work in 12 days, B in 15 days. Together?
LCM(12,15) = 60 units total work
A's rate = 60/12 = 5 units/day
B's rate = 60/15 = 4 units/day
Together = 9 units/day
Time = 60/9 = 6⅔ days
A's rate = 60/12 = 5 units/day
B's rate = 60/15 = 4 units/day
Together = 9 units/day
Time = 60/9 = 6⅔ days
Answer: 6⅔ days (or 20/3 days)
Banking: A and B together finish in 8 days. A alone in 12 days. B alone?
LCM(8,12) = 24
Together: 24/8 = 3/day
A alone: 24/12 = 2/day
B = 3 − 2 = 1/day
B alone = 24/1 = 24 days
Together: 24/8 = 3/day
A alone: 24/12 = 2/day
B = 3 − 2 = 1/day
B alone = 24/1 = 24 days
Answer: 24 days
Time, Speed, Distance — Unit Conversions
km/h to m/s: multiply by 5/18
m/s to km/h: multiply by 18/5
72 km/h = 72 × 5/18 = 20 m/s
15 m/s = 15 × 18/5 = 54 km/h
m/s to km/h: multiply by 18/5
72 km/h = 72 × 5/18 = 20 m/s
15 m/s = 15 × 18/5 = 54 km/h
Banking: A train 200m long crosses a pole in 10 seconds. Find speed in km/h.
Speed = 200/10 = 20 m/s
= 20 × 18/5 = 72 km/h
= 20 × 18/5 = 72 km/h
Answer: 72 km/h
Banking: A car covers 450 km in 6 hours. Find speed.
Speed = 450/6 = 75 km/h
Answer: 75 km/h
Banking: If A is 50% faster than B, and B takes 30 min, how long does A take?
A's speed = 1.5B. Time = Distance/Speed
A's time = B's time / 1.5 = 30/1.5 = 20 minutes
A's time = B's time / 1.5 = 30/1.5 = 20 minutes
Answer: 20 minutes
🎯 Olympiad-Level Problems
Olympiad 1: Find the last two digits of 7²⁰²³
7¹=07, 7²=49, 7³=43, 7⁴=01 (mod 100)
Cycle length = 4. 2023 mod 4 = 3
Last two digits = same as 7³ = 43
Cycle length = 4. 2023 mod 4 = 3
Last two digits = same as 7³ = 43
Answer: 43
Olympiad 2: If x + 1/x = 5, find x² + 1/x²
Square both sides: (x + 1/x)² = 25
x² + 2 + 1/x² = 25
x² + 1/x² = 23
x² + 2 + 1/x² = 25
x² + 1/x² = 23
Answer: 23
Olympiad 3: Find the sum 1² + 2² + 3² + ... + 20²
Formula: n(n+1)(2n+1)/6
= 20 × 21 × 41 / 6
= 20 × 21 × 41 / 6 = 2870
= 20 × 21 × 41 / 6
= 20 × 21 × 41 / 6 = 2870
Answer: 2,870
Practice Problems — Speed Math
P14.1
Find 125 × 88 quickly
125 × 88 = 125 × 8 × 11 = 1000 × 11 = 11,000
P14.2
A can do a job in 10 days, B in 20 days. Together?
LCM=20. A=2/day, B=1/day. Together=3/day. Time=20/3 = 6⅔ days
P14.3
Find 999 × 999 using identity
(1000−1)² = 1000000−2000+1 = 998,001
P14.4
Convert 90 km/h to m/s
90 × 5/18 = 25 m/s
P14.5
CP = ₹400, SP = ₹500. Profit %?
Profit=100. 100/400 × 100 = 25%
P14.6
Divide ₹780 in ratio 3:5:5
Total=13 parts. 780/13=60. Shares: ₹180, ₹300, ₹300
P14.7
Find 248 × 252 using identity
(250−2)(250+2) = 62500 − 4 = 62,496
P14.8
A train 300m long passes a 200m platform in 25 sec. Find speed.
Total distance = 500m. Speed = 500/25 = 20 m/s = 72 km/h
P14.9
If x + 1/x = 3, find x³ + 1/x³
x²+1/x² = 9−2=7. x³+1/x³ = (x+1/x)(x²+1/x²−1) = 3×6 = 18
P14.10
Simplify: 786 × 786 − 214 × 214
(786+214)(786−214) = 1000 × 572 = 572,000
Chapter 15
📝 Daily Practice Workbook
📚 Practice Makes Perfect
90 carefully curated problems organized by difficulty level. Work through these daily and you'll see dramatic improvement in your calculation speed. Each problem has a click-to-reveal answer!
🟢 BEGINNER Basic Operations & Tricks (30 Problems)
B1
15 × 11
165 (1_5, insert 1+5=6 in middle)
B2
35²
3×4=12, append 25. 1225
B3
10% of 870
87
B4
25% of 560
560 ÷ 4 = 140
B5
43 × 11
4_3, 4+3=7. 473
B6
75²
7×8=56, append 25. 5625
B7
50% of 990
495
B8
5% of 1,400
10%=140, half=70
B9
55²
5×6=30, append 25. 3025
B10
12 × 11
132
B11
3/4 + 1/6
(18+4)/24 = 22/24 = 11/12
B12
15% of 400
10%=40, 5%=20. 60
B13
95²
9×10=90, append 25. 9025
B14
Which is bigger: 3/8 or 5/13?
3×13=39, 5×8=40. 39<40. 5/13 is bigger
B15
∛8000
20
B16
62 × 11
6_2, 6+2=8. 682
B17
12.5% of 240
240 ÷ 8 = 30
B18
√2500
50
B19
2/5 × 15/8
Cancel: 2↔8(÷2), 5↔15(÷5). 1/1 × 3/4 = 3/4
B20
45 × 11
4_5, 4+5=9. 495
B21
33.33% of 450
450 ÷ 3 = 150
B22
85²
8×9=72, append 25. 7225
B23
Convert 0.375 to fraction
3/8
B24
75% of 360
50%=180, 25%=90. 270
B25
∛1000
10
B26
78 × 11
7_8, 7+8=15. 7(15)8 = carry: 858
B27
20% of 750
750 ÷ 5 = 150
B28
5/6 − 1/3
5/6 − 2/6 = 3/6 = 1/2
B29
√1600
40
B30
105²
10×11=110, append 25. 11025
🟡 INTERMEDIATE Nikhilam, Percentages & Fractions (30 Problems)
I1
97 × 93 (Nikhilam)
97−7, 93−7. 97−7=90 or 93−3=90. Deficit: 3,7. Left: 90, right: 21. 9021
I2
37.5% of 720
37.5% = 3/8. 720/8 × 3 = 270
I3
52² (near 50)
25+2=27, 2²=04. 2704
I4
88% of 250
90%=225, 2%=5. 88%=225−5=220
I5
∛35937
Ends in 7→root 3. 35 between 27,64→tens=3. 33
I6
104 × 96
(100+4)(100−4) = 10000−16 = 9984
I7
62.5% of 480
62.5% = 5/8. 480/8 × 5 = 300
I8
Find 44² (near 50)
25−6=19, 6²=36. 1936
I9
7/12 + 5/8
LCM=24. 14/24+15/24 = 29/24
I10
108² (near 100)
108+8=116, 8²=64. 11664
I11
Successive increases of 10%, 20% = ?% single
10+20+(10×20)/100 = 30+2 = 32%
I12
994 × 1006
(1000−6)(1000+6)=1000000−36=999,964
I13
√7056
Ends 6→root 4 or 6. Between 80²,90². Try 84: 84²=7056 ✓. 84
I14
After 25% rise, price=₹750. Original?
750 ÷ 1.25 = ₹600
I15
13³
1, 3, 9, 27. Double: 1, 6, 18, 27. Carry: 2197
I16
83² − 17²
(83+17)(83−17) = 100×66 = 6600
I17
Find 46² (near 50)
25−4=21, 4²=16. 2116
I18
87.5% of 560
87.5% = 7/8. 560/8 × 7 = 490
I19
∛91125
Ends 5→root 5. 91 between 64,125→tens=4. 45
I20
Factorize x²+8x+15
3,5. (x+3)(x+5)
I21
995²
995−5=990, 5²=025. 990025
I22
CP=₹600, 15% loss. SP?
Loss=90. SP=600−90=₹510
I23
7/9 − 2/5
(35−18)/45 = 17/45
I24
56 × 54 (base 50)
Near 50: (50+6)(50+4)=50²+50×10+24=2500+500+24=3024
I25
Convert 36 km/h to m/s
36 × 5/18 = 10 m/s
I26
A works in 6 days, B in 12 days. Together?
LCM=12. A=2/day, B=1/day. 12/3 = 4 days
I27
Find 41² (near 50 or 40)
Near 50: 25−9=16, 9²=81. 1681
I28
66.66% of 900
2/3 of 900 = 600
I29
∛46656
Ends 6→root 6. 46 between 27,64→tens=3. 36
I30
Divide ₹1200 in 2:3:7
Total=12. 1200/12=100. ₹200, ₹300, ₹700
🔴 ADVANCED Cubing, Algebra & Competition Level (30 Problems)
A1
22³ using Anurupya
8, 8, 8, 8. Double: 8, 16, 16, 8. Carry: 10648
A2
Find 113²
113+13=126, 13²=169→69 carry 1→127. 127|69. 12769
A3
786²−214²
(786+214)(786−214) = 1000×572 = 572,000
A4
If x+1/x=4, find x²+1/x²
16−2 = 14
A5
∛175616
Ends 6→root 6. 175 between 125,216→tens=5. 56
A6
Factorize 6x²+x−12
Product=−72, sum=1. Nums: 9,−8. (2x+3)(3x−4)
A7
997 × 1003
(1000−3)(1000+3) = 1000000−9 = 999,991
A8
SP=₹920 at 15% profit. Find CP.
920/1.15 = ₹800
A9
Find 89² (near 100)
89−11=78, 11²=121→21 carry 1→79. 7921
A10
Population goes up 20% then down 20%. Net?
20−20−(20×20)/100 = −4. 4% decrease
A11
17³
1, 7, 49, 343. Double: 1, 14, 98, 343. Carry: 4913
A12
√12321
Ends 1→root 1 or 9. 1|23|21. 10²<123<12². Try 111: 111²=12321 ✓. 111
A13
Simplify 437×437+563×563+2×437×563
(437+563)² = 1000² = 1,000,000
A14
A 30% then 20% discount = ? single
30+20−(30×20)/100 = 50−6 = 44%
A15
∛238328
Ends 8→root 2. 238 between 216,343→tens=6. 62
A16
Find (a+b)² if a²+b²=41, ab=20
a²+2ab+b² = 41+40 = 81
A17
1²+2²+3²+...+15²
15×16×31/6 = 1240
A18
A, B, C work together. A=10d, B=15d, C=30d. Together?
LCM=30. 3+2+1=6/day. 30/6=5 days
A19
999²
(1000−1)² = 1000000−2000+1 = 998,001
A20
Mark-up 50%, discount 20%. Profit%?
CP=100, MP=150, SP=120. 20% profit
A21
Expand (2x−3)³
8x³ − 36x² + 54x − 27
A22
Train 150m at 54 km/h. Time to cross pole?
54 km/h = 15 m/s. Time=150/15=10 sec
A23
If 73²=5329, find 730²
Add two zeros: 532,900
A24
∛389017
Ends 7→root 3. 389 between 343,512→tens=7. 73
A25
Simplify: 56²−44²
(56+44)(56−44) = 100×12 = 1200
A26
Find x³+1/x³ if x+1/x=2
x²+1/x²=4−2=2. x³+1/x³=(x+1/x)(x²+1/x²−1)=2×1=2
A27
3 consecutive 10% discounts = ?% single
0.9³ = 0.729. Discount = 1−0.729 = 27.1%
A28
18³
1,8,64,512. Double: 1,16,128,512. Carry: 5832
A29
If A:B:C = 2:3:5, B's share of ₹4000?
3/10 × 4000 = ₹1200
A30
Find 199²
(200−1)² = 40000−400+1 = 39,601
Chapter 16
🧠 Mental Math Championship
🏆 The Ultimate Challenge
You've learned all the techniques. Now it's time to put them to the ultimate test! Speed tests, brain gym puzzles, and a final championship quiz await. Can you earn the title of Vedic Math Champion?
⚡ Speed Test 1 — Addition & Subtraction
Try to solve each in under 10 seconds!
S1.1
999 + 888 + 777
(1000+900+800) − (1+12+23) = 2700−3 = 2664
S1.2
10000 − 6789
Nikhilam: 3|2|1|1 = 3211
S1.3
456 + 544
1000
S1.4
7777 − 2345
5432
S1.5
1234 + 4321
5555
S1.6
9999 − 3456
Nikhilam from 9999: 6|5|4|3 = 6543
S1.7
678 + 322 + 456 + 544
1000 + 1000 = 2000
S1.8
5000 − 1867
3133
S1.9
2468 + 7532
10000
S1.10
11111 − 9876
1235
⚡ Speed Test 2 — Multiplication
S2.1
25 × 44
25×4=100, ×11=1100. Or 25×44=1100
S2.2
99 × 47
(100−1)×47 = 4700−47 = 4653
S2.3
125 × 32
125×8=1000, 32/8=4. 1000×4=4000
S2.4
48 × 52
(50−2)(50+2)=2500−4=2496
S2.5
15 × 15
1×2=2, append 25. 225
S2.6
98 × 102
(100−2)(100+2) = 10000−4 = 9996
S2.7
75 × 75
7×8=56, append 25. 5625
S2.8
999 × 5
(1000−1)×5 = 5000−5 = 4995
S2.9
67 × 63
(65−2)(65+2)=4225−4=4221
S2.10
11 × 11 × 11
121 × 11 = 1331
⚡ Speed Test 3 — Division & Percentages
S3.1
25% of 888
888 ÷ 4 = 222
S3.2
7200 ÷ 25
7200 × 4 / 100 = 288
S3.3
12.5% of 960
960 ÷ 8 = 120
S3.4
3600 ÷ 125
3600 × 8 / 1000 = 28.8
S3.5
50% of 777
388.5
S3.6
4800 ÷ 16
300
S3.7
33.33% of 666
666 ÷ 3 = 222
S3.8
10000 ÷ 125
10000 × 8 / 1000 = 80
S3.9
75% of 440
50%=220, 25%=110. 330
S3.10
625 ÷ 25
25
🧩 Brain Gym — Pattern Recognition
BG1
What comes next: 2, 6, 12, 20, 30, ?
Pattern: n(n+1). 6×7 = 42
BG2
What comes next: 1, 1, 2, 3, 5, 8, 13, ?
Fibonacci! 8+13 = 21
BG3
What comes next: 1, 4, 9, 16, 25, ?
Perfect squares! 6² = 36
BG4
What comes next: 1, 8, 27, 64, 125, ?
Perfect cubes! 6³ = 216
BG5
If 11×11=121, 111×111=12321, then 1111×1111=?
1234321 — palindrome pattern!
🏆 Champion Challenge — Extremely Tough!
🔥 Can You Solve These?
Only true Vedic Math champions can crack these without a calculator!
CC1
Find 99999 × 99999
(100000−1)² = 10000000000−200000+1 = 9,999,800,001
CC2
Find ∛912673
Ends 3→root 7. 912 between 729(=9³) and 1000(=10³). Tens=9. 97
CC3
Find 998² + 4 × 998 + 4 (hint: identity)
(998+2)² = 1000² = 1,000,000
CC4
If x² + y² = 25 and xy = 12, find (x+y)² and (x−y)²
(x+y)² = 25+24 = 49 → x+y=7
(x−y)² = 25−24 = 1 → x−y=1
(x−y)² = 25−24 = 1 → x−y=1
CC5
Find: 1³+2³+3³+...+10³
= [10×11/2]² = 55² = 3025. (Sum of cubes = square of sum!)
🎯 Final Championship Quiz — 10 MCQs
Test your mastery of ALL Part 2 chapters. Choose wisely!
QUIZ Q1
15% of 840 = ?
QUIZ Q2
48² = ?
QUIZ Q3
∛42875 = ?
QUIZ Q4
67² − 33² = ?
QUIZ Q5
Which fraction is larger: 5/11 or 3/7?
QUIZ Q6
Successive discounts of 20% and 10% equal what single discount?
QUIZ Q7
104² = ?
QUIZ Q8
If A can do work in 12 days and B in 15 days, together they finish in:
QUIZ Q9
If x + 1/x = 5, then x² + 1/x² = ?
QUIZ Q10
1³ + 2³ + 3³ + ... + 10³ = ?
🎯 Championship Score
🏆
Congratulations!
You have completed
Vedic Mathematics Secrets — Part 2!
You've mastered Percentages, Fractions, Squares, Cubes, Algebra, Speed Math, and conquered the Mental Math Championship!
From ancient Vedic sutras to modern competitive exams — you now hold the keys to mathematical mastery. Go forth and calculate with confidence! 🌟
— EduArtha Vedic Math Academy —
"The universe is a grand book written in the language of mathematics." — Galileo Galilei
"Vedic Mathematics is a gift to the world — a system so powerful, so elegant, that it makes complex calculations feel like a game." — Sri Bharati Krishna Tirthaji
"Vedic Mathematics is a gift to the world — a system so powerful, so elegant, that it makes complex calculations feel like a game." — Sri Bharati Krishna Tirthaji