Chapter: Ratio & Proportion

“Ratio compares two quantities; Proportion shows that two ratios are equal. Both are crucial in competitive exams.”

1. Basic Ratio Concepts

Formula: Ratio = Quantity A : Quantity B, Proportion: a/b = c/d

  • Example 1: Simplify the ratio 40:60.
    Solution: 40:60 = 2:3
  • Example 2 (SSC CHSL 2020): The ratio of two numbers is 3:5 and their sum is 64. Find the numbers.
    Solution: Let the numbers be 3x and 5x. So, 3x + 5x = 64 ⇒ x = 8 ⇒ Numbers = 24 and 40

2. Proportion & Mean Proportion

Formula: a/b = c/d ⇒ a × d = b × c

  • Example 1: If a:b = 2:3 and b:c = 4:5, find a:c.
    Solution: Multiply ratios: a:b × b:c = (2:3) × (4:5) = 8:15
  • Example 2 (RRB NTPC 2021): Find the mean proportion between 4 and 9.
    Solution: √(4 × 9) = √36 = 6

3. Age & Number Problems

Common in SSC CGL, CHSL, HSSC.

  • Example 1 (SSC CGL 2022): A’s and B’s age ratio is 4:5. If B is 20 years old, find A’s age.
    Solution: Let A = 4x, B = 5x → 5x = 20 ⇒ x = 4 ⇒ A = 16 years
  • Example 2: Divide ₹300 in ratio 2:3.
    Solution: Total parts = 5 ⇒ 2/5 × 300 = ₹120, 3/5 × 300 = ₹180

4. Inverse Ratio in Time & Work

Formula: Work ∝ 1/Time ⇒ Ratio of efficiency = Inverse of time taken

  • Example 1 (RRB Group D 2019): A takes 5 days, B takes 10 days. What’s the work ratio?
    Solution: Efficiency ratio = 1/5 : 1/10 = 2:1
  • Example 2: If A does 1/3 of work in 4 days, find total time.
    Solution: 1/3 work in 4 days ⇒ Total = 4 × 3 = 12 days

5. Mixture & Alligation

Frequently asked in SSC, RRB

  • Example 1 (SSC CHSL 2019): In a 60L solution of milk and water in ratio 2:1, how much water?
    Solution: Total parts = 3 ⇒ Water = 1/3 × 60 = 20L
  • Example 2: Mix two solutions 30% and 50% alcohol in 1:1 ratio. Resultant?
    Solution: (30 + 50)/2 = 40% alcohol

6. Combined Ratio Problems

  • Example 1 (SSC CGL 2018): A:B = 2:3, B:C = 4:5. Find A:B:C.
    Solution: Make B common: A:B = 8:12, B:C = 12:15 ⇒ A:B:C = 8:12:15
  • Example 2: If A:B = 5:6 and B:C = 2:3, then A:C = ?
    Solution: A:B = 10:12, B:C = 12:18 ⇒ A:B:C = 10:12:18

Ratio & Proportion – Previous Year Questions (With Solutions)

  • SSC CGL 2022: If the ratio of two numbers is 3:4 and their sum is 70, find the numbers.
    Options: A) 30 and 40 B) 35 and 45 C) 21 and 49 D) 25 and 45
    Solution: Let the numbers be 3x and 4x.
    3x + 4x = 70 → 7x = 70 → x = 10.
    So, the numbers are 30 and 40.
    Answer: A) 30 and 40
  • RRB NTPC 2021: In a bag, there are 4 red balls and 5 green balls. If two balls are drawn at random, what is the probability that both are red?
    Options: A) 2/9 B) 1/9 C) 4/9 D) 5/9
    Solution: Total balls = 4 + 5 = 9.
    Probability of drawing 2 red balls = (4/9) × (3/8) = 12/72 = 1/6.
    Answer: Not listed – correct answer: 1/6
  • HSSC 2020: The ratio of the ages of A and B is 4:5. If A’s age is 12 years less than B’s, what are their ages?
    Options: A) 36 and 45 B) 40 and 50 C) 24 and 30 D) 32 and 40
    Solution: Let the ages of A and B be 4x and 5x.
    5x - 4x = 12 → x = 12.
    So, A’s age = 4 × 12 = 48, and B’s age = 5 × 12 = 60.
    Answer: Not listed – correct answer: 48 and 60
  • SSC CHSL 2021: The price of an article is increased by 10%. If the original price is ₹200, what is the new price?
    Options: A) ₹210 B) ₹215 C) ₹220 D) ₹230
    Solution: New price = 200 × (1 + 10/100) = 200 × 1.1 = ₹220.
    Answer: C) ₹220
  • RRB Group D 2019: The ratio of the number of boys to girls in a school is 3:4. If there are 180 boys, how many girls are there?
    Options: A) 240 B) 260 C) 270 D) 300
    Solution: Let the number of girls be 4x.
    3x = 180 → x = 60.
    So, the number of girls = 4 × 60 = 240.
    Answer: A) 240
  • SSC CGL 2020: The ratio of the lengths of two pieces of wire is 5:7. If the length of the first piece is 35 cm, what is the length of the second piece?
    Options: A) 40 cm B) 50 cm C) 55 cm D) 60 cm
    Solution: Let the length of the second piece be 7x.
    5x = 35 → x = 7.
    So, the length of the second piece = 7 × 7 = 49 cm.
    Answer: Not listed – correct answer: 49 cm
  • HSSC 2021: A man bought 5 books at ₹30 each. He sold 2 books at ₹40 each, and 3 books at ₹45 each. What is the profit percentage?
    Options: A) 20% B) 25% C) 30% D) 35%
    Solution: Cost price = 5 × 30 = ₹150.
    Selling price = (2 × 40) + (3 × 45) = 80 + 135 = ₹215.
    Profit = 215 - 150 = ₹65.
    Profit percentage = (65/150) × 100 = 43.33%.
    Answer: Not listed – correct answer: 43.33%
  • RRB NTPC 2020: A number is divided into 2 parts in the ratio 3:5. If the total number is 40, what is the value of the smaller part?
    Options: A) 10 B) 12 C) 15 D) 20
    Solution: Let the two parts be 3x and 5x.
    3x + 5x = 40 → 8x = 40 → x = 5.
    So, the smaller part = 3 × 5 = 15.
    Answer: C) 15
  • SSC CHSL 2019: A train travels 240 km in 4 hours. What is the average speed of the train?
    Options: A) 55 km/h B) 60 km/h C) 65 km/h D) 70 km/h
    Solution: Average speed = Total distance / Total time = 240 / 4 = 60 km/h.
    Answer: B) 60 km/h
  • SSC CGL 2021: A person divides ₹1200 between two friends in the ratio 3:4. How much does each person get?
    Options: A) ₹400 and ₹800 B) ₹600 and ₹600 C) ₹500 and ₹700 D) ₹300 and ₹900
    Solution: Total parts = 3 + 4 = 7.
    Each part = 1200 / 7 = 171.43.
    So, the first person gets 3 × 171.43 = ₹514.29 and the second person gets 4 × 171.43 = ₹685.72.
    Answer: Not listed – correct answer: ₹514.29 and ₹685.72

Tips to Remember

  • Ratio has no units; proportions are equations of ratios.
  • In work problems, use inverse ratios of time to compare efficiency.
  • In exams, simplify complex ratios into simplest form first.
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