Simplification / BODMAS

1. Ye topic hai kya?

Simplification = mixed expression ko ek number tak giraana. Paper likhta hai 5 + 5 × 5 − 5 ÷ 5 ya 36 ÷ 6 of 2 ya 15² − 5 × 9. Sawal yeh nahi ki “15 ka square kya hai” alag se — sawal yeh hai ki kaun pehle chalega. Galat order = galat option, jo hamesha options mein rakha hota hai.

Paper yahan kya maangta

VBODMAS, of vs ×, nested () [] {}, vinculum (bar), 1–25 squares / 1–12 cubes yaad, (a±b)² se 99², a²−b² se 51×49, simple indices, √ / ³√ of perfect, fraction-of-fraction.

Yahan kya nahi

Place / binary / unit-cycle / 3-se-katna — Number system. Bells, tiles, rem 3 — HCF / LCM. √2 proof — Real numbers. 25% of 80 — Percentage agla. Agla term 4, 9, 14… — Number series.

Indian paper: V → B → O → DM → AS. Vinculum pehle, phir brackets, phir of, phir ÷ aur × left se right, phir + aur − left se right. Calculator of nahi jaanta — isliye 12 wala trap.

2. VBODMAS — kyun yahi order

Ek expression ek sentence hai. “3 × 4” ek packet hai (12). Us packet ko 2 se jodna = 14. Agar pehle 2+3=5, phir ×4, sentence toot gaya. Isliye × ÷, + − se upar baithte hain. Same floor pe jo do kaam hon (dono ×÷, ya dono +−), left se right — kyunki 36÷12÷3 matlab pehle 36÷12=3, phir 3÷3=1, nahi 36÷(12÷3)=9.

Upar se neeche. Same step pe do signs → left se right. of ÷× se pehle: 36 ÷ 6 of 2 = 36 ÷ 12, calculator 12 nahi.

Vinculum

Bar = hidden bracket. 8 − 3 + 2 = 8 − (3+2) = 3. Bar ke bina 8 − 3 + 2 = 5 + 2 = 7 (minus phir plus, left se right: 8−3=5, 5+2=7). CSS yahan overline se bar dikhati hai — copy-paste mein bar miss ho to options 3 vs 7 dono baithe honge.

Brackets — andar se bahar

Teen shapes paper mein nested: pehle ( ), phir { }, phir [ ]. Kaam innermost se. Har bracket ke andar bhi VBODMAS chalta — [5 + 3 × 2] = [5+6]=11, nahi [8]×2.

8 − [ 6 − { 4 − (2 − 1) } ]. Sabse andar 2−1=1; phir 4−1=3; phir 6−3=3; bahar 8−3=5.

`of` vs × vs ÷

of bhi multiply hai, lekin Indian BODMAS mein ÷ aur × se pehle. Isliye:

36 ÷ 6 of 2 → pehle 6 of 2 = 12 → 36÷12 = 3.
Calculator / left-to-right ×: 36÷6=6, 6×2=12. Paper 12 ko trap rakhta hai.
2/3 of 3/4 of 96 = (2/3)×(3/4)×96. Beech ka 3 cancel: (2/4)×96 = 48.
“25% of 80” mein of hai, lekin % = /100 agle chapter. Yahan fraction-of.

36 ÷ 12 ÷ 3 = (36÷12)÷3 = 1. 36 ÷ (12÷3) = 9 — yeh tabhi jab bracket ho. 48 ÷ 12 × 4 = 4×4 = 16, nahi 48÷48=1.

Example

2 + 3 × 4 = ?

Solution

× pehle
3×4=12. Phir 2+12=14. (2+3)×4=20 trap.

Answer14

Example

48 ÷ 12 × 4 + 3 = ?

Solution

÷ × left→right
48÷12=4, 4×4=16. Phir +3=19. Agar pehle 12×4=48, 48÷48=1, +3=4 — galat floor.

Answer19

Example

36 ÷ 6 of 2 = ?

Solution

of pehle
6 of 2 = 12. 36÷12=3. Calculator 12 mat. Number system 15² yahi BODMAS — table agle section.

Answer3

Example

5 + 5 × 5 − 5 ÷ 5 = ?

Solution

× aur ÷
5×5=25, 5÷5=1. Same floor, left se right — yahan alag jagah, order same.
+ −
5+25−1=29. Chain ((5+5)×5−5)÷5=9 trap. 5+5=10×5=50−1=49 bhi trap.

Answer29

Example

8 − [6 − {4 − (2 − 1)}] = ?

Solution

( )
2−1=1.
{ }
4−1=3.
[ ]
6−3=3. Bahar 8−3=5. Signs ka minus-minus + mat banao jab tak bracket na khule.

Answer5

Example

8 − 3 + 2 vs 8 − 3 + 2.

Solution

Bar
8−(3+2)=3.
Bina bar
8−3+2=7. Do alag sawal.

Answer3 with bar; 7 without

3. Squares, cubes, roots

LDC mein time bachaane ke liye 1–25 squares aur 1–12 cubes yaad. Unit-digit cycle (7²³ ka last digit) Number system pe — yahan poora value, expression ke andar.

(n+1)² = n² + 2n + 1. Jaise 26² = 25² + 50 + 1 = 625+51 = 676. Cube: (n+1)³ = n³ + 3n² + 3n + 1 — paper kam maangta; 11³=1331, 12³=1728 likh ke rakho.

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	30	900
7	49	17	289
8	64	18	324
9	81	19	361
10	100	20	400

n	1	2	3	4	5	6	7	8	9	10	11	12
n³	1	8	27	64	125	216	343	512	729	1000	1331	1728

15² = 225 — Number system ne yahan bheja tha. 8²+15²=17² triangles wala triple yahan mat ghusaana. √4=2 rational Real numbers; yahan √256=16, ³√64=4, √0.16=0.4, ³√0.008=0.2 (kyunki 0.2×0.2×0.2=0.008).

√(ab)=√a × √b jab a,b ≥ 0. Isliye √12 = √(4×3)=2√3. √12 + √27 = 2√3 + 3√3 = 5√3. √12 + √27 = √39 nahi — alag radicals tab add jab same andar.

Example

15² − 5 × 9 + 12 ÷ 4 = ?

Solution

Powers / ×÷
15²=225. 5×9=45. 12÷4=3.
+ − left
225 − 45 + 3 = 183. (15²−5)×9 mat. 225−(5×9+12)÷4 mat.

Answer183

Example

√256 + ³√64 + √0.16 = ?

Solution

Table
16²=256 → 16. 4³=64 → 4. 0.4²=0.16 → 0.4. Sum 20.4.

Answer20.4

4. Identities, indices, fractions

Polynomials wala remainder / factor theorem yahan nahi. Yahan identities number ko shortcut:

(a+b)² = a²+2ab+b²
(a−b)² = a²−2ab+b²
a²−b² = (a−b)(a+b)
99² = (100−1)² = 10000 − 200 + 1 = 9801. 101² = 10201. 51×49 = (50+1)(50−1) = 2500−1 = 2499.

Indices (same base, numbers):

a^m × aⁿ = a^m+n. a^m / aⁿ = a^m−n.
(a^m)ⁿ = a^mn. (ab)ⁿ = aⁿbⁿ.
a⁰ = 1 (a ≠ 0). a⁻ⁿ = 1/aⁿ.
√a = a^1/2. ³√a = a^1/3.

Example: 2⁵ × 4³ / 8². 4=2², 8=2³ → 2⁵ × (2²)³ / (2³)² = 2⁵ × 2⁶ / 2⁶ = 2⁵ = 32.

2³ × 2³ = 2⁶=64, nahi 2⁹. (2³)³=2⁹=512. a^m + aⁿ ko a^m+n mat. Binary 101101 Number system. 2ⁿ unit cycle bhi wahan.

Fractions: 1/2 + 1/3 + 1/6. LCM of dens 6 — yeh addition ka LCM, bells wala “saath 240 s” nahi. 3/6+2/6+1/6=1. 0.25×0.25=0.0625 (25×25=625, do decimal × do decimal = 4 digits: 0.0625).

Example

99² = ?

Solution

(100−1)²
10000 − 2×100×1 + 1 = 9801. 99×99 lambi multiply mat.

Answer9801

Example

51 × 49 = ?

Solution

a²−b²
(50+1)(50−1)=50²−1²=2500−1=2499.

Answer2499

Example

2/3 of 3/4 of 96 = ?

Solution

of = ×
(2/3)×(3/4)×96. 3 cancel. (2/4)×96=48. 25% of 96 Percentage pe.

Answer48

Example

√12 + √27 = ?

Solution

Factor
√12=2√3, √27=3√3. Sum 5√3. √39 nahi.

Answer5√3

5. Sawal — basic se pro

Q1 · Basic

2 + 3 × 4 = ?

Solution

× pehle
12+2=14. B (2+3)×4.

Answer14

Q2 · Basic

48 ÷ 12 × 4 = ?

Solution

Left→right
4×4=16. B 48÷48.

Answer16

Q3 · Basic

36 ÷ 6 of 2 = ?

Solution

of
36÷12=3. B calculator.

Answer3

Q4 · Medium

5 + 5 × 5 − 5 ÷ 5 = ?

Solution

×÷ then +−
5+25−1=29. B nested-all. D 5+5×5−1 skip ÷ place.

Answer29

Q5 · Medium

15² − 5 × 9 + 12 ÷ 4 = ?

Solution

225−45+3
183. Number system table yahin. Triangles 8-15-17 nahi.

Answer183

Q6 · Medium

8 − [6 − {4 − (2 − 1)}] = ?

Solution

Andar se
1 → 3 → 3 → 8−3=5.

Answer5

Q7 · Medium

99² = ?

9801
9901
9999
9701

Solution

(100−1)²
10000−200+1=9801. B 10000−99.

Answer9801

Q8 · Pro

√256 + ³√64 = ?

Solution

16+4
20. B 16+2 (√64 mix). C ³√256 mix.

Answer20

Q9 · Pro

2/3 of 3/4 of 96 = ?

Solution

Cancel 3
(2/4)×96=48. % nahi.

Answer48

Q10 · Pro

Bells 16, 20, 24 s saath. Is chapter?

Nahi — HCF / LCM
Haan, LCM BODMAS
15² table
of = 240

Solution

Saath = LCM
A. 240 s wahan. Yahan order of operations.

AnswerA

Practice

Pehle khud, phir neeche kholo.

1 2 3 4 5 6 7 8 9 10

36 ÷ 12 ÷ 3 = ?

Solution

Left→right
3÷3=1. B 36÷4.

Answer1

8 − 3 + 2 = ?

Solution

Bar = bracket
8−5=3. B bina bar.

Answer3

51 × 49 = ?

2499
2501
2599
2401

Solution

50²−1
2499. D 49².

Answer2499

26² = ?

Solution

25²+50+1
676. C 24².

Answer676

√12 + √27 = ?

5√3
√39
6√3
3√6

Solution

2√3+3√3
5√3. B add under root.

Answer5√3

2⁵ × 4³ / 8² = ?

Solution

Base 2
2⁵×2⁶/2⁶=32. Binary 101101 nahi.

Answer32

³√0.008 = ?

0.2
0.02
0.8
0.002

Solution

0.2³
0.008. B 0.02³=0.000008.

Answer0.2

1/2 + 1/3 + 1/6 = ?

Solution

LCM 6
3/6+2/6+1/6=1. Bells LCM 240 nahi.

Answer1

0.25 × 0.25 = ?

0.0625
0.625
0.5
6.25

Solution

625, 4 decimals
0.0625. 1/4 × 1/4 = 1/16.

Answer0.0625

P10

25% of 80. Is chapter?

Nahi — Percentage
Haan, of = BODMAS only
Binary 80
HCF 20

Solution

%
A. Fraction 1/4 of 80 yahan 20. 25% formula agla.

AnswerA

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	30	900
7	49	17	289
8	64	18	324
9	81	19	361
10	100	20	400

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	30	900
7	49	17	289
8	64	18	324
9	81	19	361
10	100	20	400

1. Ye topic hai kya?

Paper yahan kya maangta

Yahan kya nahi

2. VBODMAS — kyun yahi order

Vinculum

Brackets — andar se bahar

of vs × vs ÷

3. Squares, cubes, roots

4. Identities, indices, fractions

5. Sawal — basic se pro

Practice

`of` vs × vs ÷

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	30	900
7	49	17	289
8	64	18	324
9	81	19	361
10	100	20	400