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Ratio and proportion

Lesson 86 of 113

1. Ye topic hai kya?

Ratio a : b = a/b — do quantities ka comparison, same unit. Proportion do ratios ka barabar: a : b = c : d matlab ad = bc. Paper: simplify, teen ka chain, paisa baanto, missing mean, “2A=3B=4C”, kapda-price (direct), pumps-hours (inverse definition).

Paper yahan kya maangta

HCF se chhota ratio, duplicate/inverse/compound, chain A:B aur B:C, divide in m:n, 2A=3B=4C, mean/third/fourth proportional, componendo (a+b):(a−b), x ∝ y vs xy = k.

Yahan kya nahi

25% of 80, +20 phir −20 — Percentage. Pita–beta, n saal pehle ratio — Age. a1/a2 ≠ b1/b2 — Pair of linear. Milk–water rule — Mixture. Capital × time — Partnership. Men-days poora — Time and work. 48,180 bells — HCF/LCM.

Ratio tab likho jab unit same ho (₹:₹, cm:cm). 2 m : 50 cm pehle 200:50 = 4:1. Extremes × = means ×: a:b = c:d ⇒ ad = bc.

2. Ratio, types, chain

45:60 ko HCF 15 se kaato → 3:4. Tool HCF hai, sawal “tile/bells” nahi. Order matter: 3:4 ≠ 4:3. 0 ratio mein nahi. Negative paper mein almost nahi.

2 3

2 : 3. Pehla, doosre ka 2/3. Sum ke hisse 2/5 aur 3/5 — yeh “of the total”, Percentage wala 25% of 80 alag.

Naam2 : 3 se
Inverse3 : 2
Duplicate (squares)4 : 9
Triplicate (cubes)8 : 27
Compound with 4 : 58 : 15 (multiply same side)

Kaun bada: 3:4 vs 5:7 → cross 3×7=21, 5×4=20. 21>20 ⇒ 3/4 bada. (Cross multiply, BODMAS 15² nahi.)

Chain — beech wala same

A:B = 2:3, B:C = 4:5. B alag-alag hai (3 vs 4). LCM(3,4)=12.

  • A:B = 2:3 = 8:12
  • B:C = 4:5 = 12:15
  • A:B:C = 8:12:15
A = 8 B = 12 (common) C = 15

8 : 12 : 15. Teen numbers ka ratio — pair-linear ke a1/a2 = b1/b2 wala k yahan nahi.

A:B=2:3 aur B:C=4:5 ko seedha 2:3:5 mat. B match karna zaroori. 2:3:4:5 bhi galat (wo four terms).
Example

45 : 60 simplify.

Solution

  1. HCF 15

    3:4. 48:180 HCF chapter ka pair yahan mat.

Answer3 : 4
Example

A:B = 2:3, B:C = 4:5. A:B:C?

Solution

  1. B = 12

    8:12 aur 12:15 → 8:12:15.

Answer8 : 12 : 15
Example

2A = 3B = 4C. A:B:C?

Solution

  1. Common 12

    LCM(2,3,4)=12. 2A=12 ⇒ A=6. 3B=12 ⇒ B=4. 4C=12 ⇒ C=3. Ratio 6:4:3.

Answer6 : 4 : 3

3. Proportion — mean, third, fourth

a : b :: c : d (proportion) jab ad = bc. a,d extremes; b,c means.

Fourth proportional of a,b,c = x jahan a:b = c:x → x = bc/a.
Third of a,b = x jahan a:b = b:x → x = b²/a (continued).
Mean of a,c = b jahan a:b = b:c → b² = ac, b = √(ac) (positive).

Continued proportion a:b:c matlab a/b = b/c, ek hi mean. 3:6:12 continued (6²=3×12). 3:6:10 nahi.

Agar a/b = c/d, to:

  • Invertendo: b/a = d/c
  • Alternendo: a/c = b/d
  • Componendo: (a+b)/b = (c+d)/d
  • Dividendo: (a−b)/b = (c−d)/d
  • Componendo–dividendo: (a+b)/(a−b) = (c+d)/(c−d)
Example

Mean proportional 3 aur 12 ke beech?

Solution

  1. √(3×12)

    √36=6. Check 3:6 = 6:12. −6 paper nahi.

Answer6
Example

Fourth proportional 3, 4, 9?

Solution

  1. 3:4 = 9:x

    3x=36, x=12. Third mat (wo 4²/3).

Answer12
Example

Third proportional 4 aur 6?

Solution

  1. 4:6 = 6:x

    4x=36, x=9.

Answer9
Example

a:b = 4:3. (a+b) : (a−b)?

Solution

  1. a=4k, b=3k

    (7k):(k)=7:1. 3:4 pe a−b negative — isliye 4:3 liya. Age 3:4 umar nahi.

Answer7 : 1

4. Amount baantna, direct / inverse

₹840 ko 3:4 mein. Hisse 3+4=7. Ek hisse = 840/7=120. Pehla 360, doosra 480. Teen 2:3:5, sum 150: unit 15 → 30, 45, 75.

360 (3) 480 (4)

360+480=840. 3/7 aur 4/7. Coordinate section formula yahan nahi — quantity baantna. Umar-sum Age page.

Coins: 50-paise aur ₹1, ginti 3:2, kul 70. 3x+2x=70, x=14. 42×0.5 + 28×1 = 21+28 = ₹49. Alligation (milk %) nahi. Partnership (time×capital) nahi.

Direct ∝ aur inverse

Direct: x/y constant. Kapda 8 m = ₹160 ⇒ 1 m = 20 ⇒ 12 m = 240. (Double length, double cost.)

Inverse: xy constant. 6 pumps 8 hours (same tank) ⇒ 6×8=48. 8 pumps: 48/8=6 hours. Zyada pumps, kam time. Pipes / Time-and-work poora chapter baad mein — yahan sirf inverse ki pehchaan.

Direct ko inverse mat. Speed–time TSD pe; boats nahi. Map 1:50000 alag scale ratio, yahan numbers 8:12:15. 25% of 80 Percentage. 2+3×4 Simplification.
Example

₹840 ko 3:4 mein baanto.

Solution

  1. 7 hisse

    120×3=360, 120×4=480.

Answer360 and 480
Example

2:3:5, sum 150. Numbers?

Solution

  1. 10 hisse

    15×2,3,5 = 30, 45, 75. Check 150.

Answer30, 45, 75
Example

8 m cloth ₹160. 12 m?

Solution

  1. Direct

    160/8×12=240. Inverse 160×8/12 mat.

Answer240
Example

6 pumps, 8 h. 8 pumps, kitne h (same tank)?

Solution

  1. xy = k

    48/8=6. Time and work / pipes depth baad.

Answer6 h

5. Sawal — basic se pro

Q1 · Basic

45 : 60 = ?

  1. 3 : 4
  2. 4 : 3
  3. 45 : 60
  4. 15 : 20

Solution

  1. HCF 15

    3:4. D abhi simplify nahi. B ulta.

Answer3 : 4
Q2 · Basic

Mean proportional 3 aur 12?

  1. 6
  2. 7.5
  3. 36
  4. 4

Solution

  1. √36

    6. B average — Average agla.

Answer6
Q3 · Basic

2 : 3 = 4 : x. x?

  1. 6
  2. 5
  3. 8
  4. 1.5

Solution

  1. 2x=12

    6.

Answer6
Q4 · Medium

₹840 in 3 : 4. Chhota hissa?

  1. 360
  2. 480
  3. 280
  4. 420

Solution

  1. 3/7 × 840

    360. B bada. D half.

Answer360
Q5 · Medium

A:B=2:3, B:C=4:5. A:B:C?

  1. 8 : 12 : 15
  2. 2 : 3 : 5
  3. 2 : 4 : 5
  4. 8 : 12 : 20

Solution

  1. B=12

    8:12:15. B B-match nahi.

Answer8 : 12 : 15
Q6 · Medium

2 : 3 : 5, sum 150. Sabse bada?

  1. 75
  2. 50
  3. 90
  4. 45

Solution

  1. 5/10 × 150

    75.

Answer75
Q7 · Medium

2A = 3B = 4C. A:B:C?

  1. 6 : 4 : 3
  2. 2 : 3 : 4
  3. 3 : 4 : 6
  4. 4 : 3 : 2

Solution

  1. 12

    6:4:3. B coefficients hi ratio mat.

Answer6 : 4 : 3
Q8 · Pro

a:b = 4:3. (a+b):(a−b)?

  1. 7 : 1
  2. 7 : 4
  3. 1 : 7
  4. 4 : 3

Solution

  1. 7k : k

    7:1.

Answer7 : 1
Q9 · Pro

Duplicate of 2 : 3?

  1. 4 : 9
  2. 8 : 27
  3. 3 : 2
  4. 2 : 9

Solution

  1. Squares

    4:9. B triplicate. C inverse.

Answer4 : 9
Q10 · Pro

25% of 80. Is chapter?

  1. Nahi — Percentage
  2. Haan, 20 : 80
  3. Ratio 1 : 4 only
  4. HCF 20

Solution

  1. %

    A. 1:4 yahan tool ho sakta, sawal % ka wahan.

AnswerA

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

Fourth proportional 3, 4, 9?

  1. 12
  2. 6
  3. 16
  4. 7

Solution

  1. 3x=36

    12. B mean 3,12 mix.

Answer12
P2

Third proportional 4, 6?

  1. 9
  2. 8
  3. 10
  4. 5

Solution

  1. 4x=36

    9.

Answer9
P3

Compound 2:3 and 4:5?

  1. 8 : 15
  2. 6 : 8
  3. 8 : 12
  4. 2 : 15

Solution

  1. 2×4 : 3×5

    8:15.

Answer8 : 15
P4

Inverse of 4 : 5?

  1. 5 : 4
  2. 16 : 25
  3. 4 : 5
  4. 1 : 1

Solution

  1. Ulta

    5:4. B duplicate.

Answer5 : 4
P5

8 m = ₹160. 12 m?

  1. 240
  2. 120
  3. 192
  4. 80

Solution

  1. Direct

    240. B inverse.

Answer240
P6

6 pumps 8 h. 8 pumps?

  1. 6 h
  2. 10⅔ h
  3. 4 h
  4. 8 h

Solution

  1. 48/8

    6 h. B direct 8/6×8.

Answer6 h
P7

50p : ₹1 coins = 3:2, kul 70. Value?

  1. ₹49
  2. ₹70
  3. ₹35
  4. ₹42

Solution

  1. 42×½ + 28

    49. Mixture alligation nahi.

Answer₹49
P8

3:4 vs 5:7. Kaun bada ratio?

  1. 3 : 4
  2. 5 : 7
  3. Equal
  4. Cannot

Solution

  1. 21 vs 20

    3/4 bada.

Answer3 : 4
P9

2 : 3. Pehla, doosre ka kitna %?

  1. 66⅔%
  2. 60%
  3. 40%
  4. 150%

Solution

  1. 2/3 × 100

    66⅔%. C 2/5 of sum. % formula Percentage, yahan ratio se.

Answer66⅔%
P10

Pita–beta, 8 saal baad double. Is chapter?

  1. Nahi — Age
  2. Haan, 3 : 1
  3. Ratio 2A=3B
  4. Partnership

Solution

  1. Time-shift

    A. Umar Age pe. Yahan paisa/hisse.

AnswerA

← Index · Percentage · Simplification

Agle topic: Average.