Mixture / alligation

1. Ye topic hai kya?

Mixture = do liquids / do teas / do % solutions milana. Alligation = mean ke aas-paas shortcut: sasta kitna, mehnga kitna. Partnership P10 yahi bheja: milk–water, C×T nahi.

Paper yahan kya maangta

Do prices ka mean, (d−m):(m−c), kitna sasta jodo, milk:water badalna, % water ghatane ke liye milk, vessel se nikaal ke water bharna (1−y/x)ⁿ, do mixtures ek saath.

Yahan kya nahi

C×T, 8:10:9, working 10% — Partnership. A:B=2:3 B:C=4:5 → 8:12:15 — Ratio. CP 200 SP 250 — Profit/loss. 25% of 80 — Percentage. 6 men 8 din / A 10 din — Time and work.

Alligation: cheap : dear = (dear − mean) : (mean − cheap)
Replacement: milk left = x (1 − y/x)ⁿ — x vessel, y nikaala, n baar.

2. Alligation — kyun 2:3

₹30/kg aur ₹40/kg, mean ₹36. Farq: 40−36=4 (yeh saste ka hissa), 36−30=6 (mehnge ka). 4:6 = 2:3. Check: 2×30 + 3×40 = 60+120=180, 5 kg, 180/5=36.

Upar prices, beech mean, neeche farq. Sasta:mehnga = opposite farq. 8:12:15 nahi.

Agar mehnga 20 kg @40 pehle se hai, mean 30 chahiye, sasta @25: (40−30):(30−25)=10:5=2:1. Sasta = 2×20 = 40 kg.

% solutions bhi same: 40% aur 70% milake 50%. (70−50):(50−40)=20:10=2:1 — matlab 40% wala zyada (2 hisse), 70% ek hissa.

Ye weighted mean ka shortcut hai. Average-page ka 30 bacche @40 + 20 @50 = 44 alag story (log, mix nahi). 2xy/(x+y) speed Average pe.

Example

₹30 aur ₹40, mean ₹36. Cheap:dear?

Solution

4:6
2:3. 1:1 mat (mid 35 hota).

Answer2:3

Example

20 kg @₹40. Mean ₹30, sasta @₹25. Kitna sasta?

Solution

2:1
Sasta:dear=2:1, dear=20 → sasta=40 kg.

Answer40 kg

3. Milk–water

Pehle kitna milk, kitna water likho. Ratio milk:water. Water jodne se milk same rehta. Milk jodne se water same.

40 L milk, 4:1 chahiye → water 10 L. Milk 40 same. Partnership C×T nahi.

40 L shuddh milk, 4:1 banana: 4 parts milk = 40, 1 part water = 10 L jodo.

15 L pehle se 4:1: milk 12, water 3. Ab 3:1 chahiye: 12 : w = 3:1 → w=4. Water 1 L aur jodo (3 se 4).

36 L mixture 5:1: 6 parts=36, 1 part=6, water=6 L, milk=30.

60 L, 20% water (water 12, milk 48). Water 10% karna hai — water 12 hi rahega, milk jodo. 12 = 10% of total → total 120, extra milk = 120−60 = 60 L.

Example

40 L milk. 4:1 milk:water. Kitna water?

Solution

40/4
10 L. 8 L se 5:1 hota.

Answer10 L

Example

60 L, 20% water. Milk jodo taaki water 10% ho.

Solution

12 = 10% of T
T=120, milk add=60. 25% of 80 Percentage, yahan nahi.

Answer60 L

4. Nikaal ke water bharna, do mixes

Vessel x litre milk, y litre nikaalo, water bharo, n baar. Har baar milk ka fraction (1−y/x) reh jaata.

Milk bacha = x (1 − y/x)ⁿ
80 L, 20 L nikaala, 1 baar: 80×3/4 = 60. Do baar: 80×(3/4)² = 45.

64 L, 16 L, do baar: 64×(3/4)² = 36 L milk. (16+16=32 nikaal diya socha 32 bacha — galat, kyunki doosri baar mixture nikalta hai, shuddh milk nahi.)

Do mixtures equal volume: 3:1 aur 5:3. Milk hisse 3/4 aur 5/8. Average (6/8 + 5/8)/2 = 11/16 milk, water 5/16 → 11:5.

Alligation mein cheap:dear ko (m−c):(d−m) ulta mat. Water jodne pe milk mat badhao. Replacement pe y ko n×y seedha minus mat. 8:12:15 chain Ratio. C×T Partnership. A 10 din, B 15 din Time-and-work.

Example

80 L milk, 20 L nikaal ke water, 2 baar. Milk bacha?

Solution

(3/4)²
80×9/16=45. 40 nahi (20+20).

Answer45 L

Example

3:1 aur 5:3, equal litre mix. Naya milk:water?

Solution

11/16 : 5/16
11:5. 8:4=2:1 mat (ratios jodna).

Answer11:5

5. Sawal — basic se pro

Q1 · Basic

₹30, ₹40, mean ₹36. Cheap:dear?

Solution

4:6
2:3. B ulta.

Answer2:3

Q2 · Basic

20 kg @40. Mean 30, sasta @25. Sasta kg?

Solution

2:1
40 kg.

Answer40

Q3 · Basic

40 L milk, 4:1. Water?

10 L
8 L
4 L
40 L

Solution

10
B 5:1.

Answer10 L

Q4 · Medium

15 L of 4:1. Water jodo 3:1 ke liye.

Solution

12:3 → 12:4
+1 L.

Answer1 L

Q5 · Medium

40% aur 70%, mean 50%. Ratio (40% : 70%)?

Solution

20:10
2:1. B ulta (70 ke paas socha).

Answer2:1

Q6 · Medium

36 L mixture 5:1. Water kitna?

6 L
5 L
30 L
7.2 L

Solution

36/6
6. C milk.

Answer6 L

Q7 · Medium

80 L, 20 L replace, 2 baar. Milk left?

45 L
40 L
60 L
48 L

Solution

80×9/16
45. B 20+20. C ek baar.

Answer45 L

Q8 · Pro

60 L, 20% water. Milk add, water 10%. Kitna milk?

60 L
12 L
48 L
6 L

Solution

12 / 0.1 − 60
60. D 25% of 80 mix.

Answer60 L

Q9 · Pro

3:1 aur 5:3 equal mix. Milk:water?

11:5
8:4
15:7
4:3

Solution

11/16
11:5. B ratios jodna.

Answer11:5

Q10 · Pro

Profit share capital × months. Is chapter?

Nahi — Partnership
Haan, alligation 8:10:9
Milk C×T
2:3 partners

Solution

Firm
A. 8:10:9 wahan C×T.

AnswerA

Practice

Pehle khud, phir neeche kholo.

1 2 3 4 5 6 7 8 9 10

₹20 aur ₹50, mean ₹30. Cheap:dear?

2:1
1:2
20:50
3:2

Solution

20:10
2:1.

Answer2:1

70 L mixture 5:2 milk:water. Water?

20 L
50 L
14 L
35 L

Solution

2/7 × 70
20. B milk.

Answer20 L

40 L milk + 8 L water. Ratio?

5:1
4:1
8:40
5:2

Solution

40:8
5:1.

Answer5:1

80 L milk, 20 L replace, 1 baar. Milk left?

60 L
45 L
20 L
80 L

Solution

×3/4
60. B do baar.

Answer60 L

64 L, 16 L replace, 2 baar. Milk left?

36 L
32 L
48 L
40 L

Solution

(3/4)² × 64
36. B 16+16.

Answer36 L

Tea ₹25 aur ₹40, mix 2:3. Mean ₹?

34
32.5
30
36

Solution

(50+120)/5
34. D 30-40 wala 36.

Answer34

10% aur 40% acid, mean 25%. Ratio?

1:1
1:2
2:1
10:40

Solution

15:15
1:1.

Answer1:1

A:B:C = 8:10:9 from C×T. Is chapter?

Nahi — Partnership
Haan, alligation 8:10:9
Milk 8:10:9
8:12:15

Solution

Months
A.

AnswerA

A:B=2:3, B:C=4:5 → 8:12:15. Is chapter?

Nahi — Ratio
Haan, mix 8:12:15
Alligation chain
Partnership

Solution

Chain
A. Wahan 8:12:15.

AnswerA

P10

A 10 din, B 15 din, saath kitne din. Is chapter?

Nahi — Time and work
Haan, mix 2:3
Alligation days
Pipes

Solution

Kaam
A. 1/10+1/15 agle pe. Pipes uske baad.

AnswerA