📘 Ratio and Proportion – Important Notes
🔹 Ratio
- A ratio is the comparison of two quantities of the same kind.
- Example: Apples : Oranges = 2 : 3
- General Form: ab
✅ Key Points
- Ratios are dimensionless.
- Multiplying or dividing both terms by the same number does not change the ratio.
- Example: 2:3 = 4:6 = 6:9
🔹 Proportion
- A proportion shows equality of two ratios.
- Example: 23 = 46
✅ Rule of Proportion
🔹 Types of Proportion
- Direct Proportion: Cost ∝ Quantity
- Inverse Proportion: Speed ∝ 1Time
- Continued Proportion: If a:b = b:c, then b is mean proportional.
🔹 Important Formulas
- If a:b = c:d ⇒ a × d = b × c
- Mean Proportional = √(a × b)
- Third Proportional = b²a
- Fourth Proportional = b × ca
🔹 Detailed Concepts
Antecedent & Consequent:
In x : y → x = antecedent, y = consequent.
Extremes & Means:
If a : b :: c : d
Extremes = a, d | Means = b, c
👉 a × d = b × c
First Proportional: If x : a :: b : c
Fourth Proportional: If a : b :: c : x
Third Proportional: If a : b :: b : x
Questions — Ratio & Proportion
Q1.
Question: If a : b = 7 : 9 and b : c = 15 : 7, then what is a : c?
Options:
(1) 5 : 3
(2) 3 : 5
(3) 7 : 21
(4) 7 : 15
Show Answer
Correct Option: (1) 5 : 3
Explanation: Make middle terms equal. LCM of 9 and 15 = 45. Convert: a:b = 35:45 and b:c = 45:21. So a:b:c = 35:45:21 ⇒ a:c = 35:21 = 5:3.
Q2.
Question: If A : B = 6 : 8 and B : C = 5 : 12, then find A : B : C.
Options:
(1) 6 : 8 : 12
(2) 15 : 20 : 48
(3) 3 : 4 : 9
(4) 30 : 40 : 96
Show Answer
Correct Option: (2) 15 : 20 : 48
Explanation: Make B same: LCM(8,5)=40. Scale A:B = 30:40 and B:C = 40:96 ⇒ A:B:C = 30:40:96 ⇒ divide by 2 ⇒ 15:20:48.
Q3.
Question: Rs. 6200 is divided into three parts proportional to 1/2 : 1/3 : 1/5. The three parts are:
Options:
(1) 3000, 2000, 1200
(2) 3500, 1500, 1200
(3) 2500, 2000, 1700
(4) 2200, 3000, 1000
Show Answer
Correct Option: (1) 3000, 2000, 1200
Explanation: Convert fractions to whole-number ratio. Multiply each by 30: 1/2→15, 1/3→10, 1/5→6. Sum=31. One part=200. So shares=3000, 2000, 1200.
Q4.
Question: The third proportional to 0.8 and 0.2 is:
Options:
(1) 0.05
(2) 0.8
(3) 0.4
(4) 0.032
Show Answer
Correct Option: (1) 0.05
Explanation: x = b²/a = 0.2² / 0.8 = 0.05.
Q5.
Question: What must be added to each term of the ratio 2 : 5 so that it becomes 5 : 6?
Options:
(1) 65
(2) 78
(3) 13
(4) 12
Show Answer
Correct Option: (3) 13
Explanation: (2+x)/(5+x) = 5/6 ⇒ x=13.
Q6.
Question: In a college union there are 48 students. The ratio of boys to girls is 5 : 3. The number of girls to be added so that the ratio of boys to girls becomes 6 : 5 is:
Options:
(1) 6
(2) 7
(3) 12
(4) 17
Show Answer
Correct Option: (2) 7
Explanation: Boys=30, girls=18. Let added girls=x. 30:(18+x)=6:5 ⇒ x=7.
Q7.
Question: Rs. 782 is divided into three parts in the ratio 1/2 : 2/3 : 3/4. The first part is:
Options:
(1) Rs. 182
(2) Rs. 204
(3) Rs. 190
(4) Rs. 196
Show Answer
Correct Option: (2) Rs. 204
Explanation: Convert: 6:8:9. Sum=23. One part=34. First=6×34=204.
Q8.
Question: The lengths of the sides of a triangle are in the ratio 9 : 4 : 3. If the perimeter is 720 cm, find the sides.
Show Answer
Correct Option: 405, 180, 135
Explanation: Total parts=16. One part=45. Sides=405,180,135.
Q9.
Question: The ratio of two numbers is 3 : 8 and their difference is 115. The smaller number is:
Options: (1) 184 (2) 194 (3) 69 (4) 59
Show Answer
Correct Option: (3) 69
Explanation: 5k=115 ⇒ k=23 ⇒ smaller=69.
Q10.
Question: The sum of two numbers is 40 and their difference is 4. The ratio of the numbers is:
Options: (1) 21:19 (2) 22:9 (3) 11:9 (4) 11:18
Show Answer
Correct Option: (3) 11 : 9
Explanation: Numbers=22,18 ⇒ ratio=11:9.
Q11.
Question: Two numbers are in the ratio 1 : 3. If their sum is 240, then their difference is:
Options: (1) 120 (2) 108 (3) 100 (4) 96
Show Answer
Correct Option: (1) 120
Explanation: Numbers=60,180 ⇒ difference=120.
Q12.
Question: Two numbers are in the ratio 3 : 5. If 9 is subtracted from each, they become 12 : 23. Find the smaller number.
Options: (1) 27 (2) 33 (3) 49 (4) 55
Show Answer
Correct Option: (2) 33
Explanation: 3k−9 : 5k−9 = 12:23 ⇒ k=11 ⇒ smaller=33.
Q13.
Question: A mixture contains spirit and water in ratio 3 : 2. If spirit is 3 litres more than water, the spirit quantity is:
Options: (1) 10 (2) 12 (3) 8 (4) 9 litres
Show Answer
Correct Option: (4) 9 litres
Explanation: 3k−2k=3 ⇒ k=3 ⇒ spirit=9 litres.
Q14.
Question: Boys:girls=2:1 after 15 girls leave. Then after 45 boys leave, ratio=1:5. Originally, girls= ?
Options: (1)20 (2)30 (3)40 (4)50
Show Answer
Correct Option: (3) 40
Explanation: Equations ⇒ G=40.
Q15.
Question: Find the fourth proportional to 18, 36, 52.
Options: (1)36 (2)52 (3)104 (4)18
Show Answer
Correct Option: (3) 104
Explanation: x=36×52/18=104.
Q16.
Question: Find the third proportional to 16 and 24.
Options: (1)36 (2)24 (3)48 (4)16
Show Answer
Correct Option: (1) 36
Explanation: x=24²/16=36.
Q17.
Question: Radhika got coins worth ₹1845 in ratio 3:5:7 (₹5, ₹10, ₹20). What is worth of ₹5 coins?
Options: (1)27 (2)135 (3)405 (4)9
Show Answer
Correct Option: (2) ₹135
Explanation: 205k=1845 ⇒ k=9 ⇒ 5-rupee coins=27 ⇒ value=135.
Q18.
Question: There are ₹225 consisting of 1-rupee, 50p, 25p coins in ratio 8:5:3. Find number of 1-rupee coins.
Options: (a)80 (b)112 (c)160 (d)172
Show Answer
Correct Option: (c) 160
Explanation: Total paise=22500. Value ratio=1125k ⇒ k=20 ⇒ 1-rupee coins=160.
Q19.
Question: A box has 280 coins of 1-rupee, 50p, 25p. Their values are in ratio 8:4:3. Number of 50p coins?
Options: (a)70 (b)60 (c)80 (d)90
Show Answer
Correct Option: (c) 80
Explanation: Solve values ⇒ t=1000 ⇒ n2=80.
📘 Ratio and Proportion – Video Classes
