Ratio and Proportion: Notes और MCQ Practice

आसान भाषा में: Ratio का मतलब दो quantities का relative size, और Proportion का मतलब दो ratios का बराबर होना।

Important Formulas / Rules

• Ratio = a : b = a ÷ b
• If a:b = c:d then a × d = b × c (cross‑multiplication)
• Direct proportion: y = k × x
• Percentage change = (New − Old) ÷ Old × 100

Quick Tricks

1. दोनों terms को उनके HCF से divide करो → simplest ratio.
2. Direct proportion में, अगर एक quantity double हो तो दूसरा भी double हो जाता है.

Common Mistakes

1. Ratio को fraction समझना – a:b ≠ a/b unless you convert.
2. Units को ignore करना – सभी quantities का unit समान होना चाहिए.

1. Concept

Ratio is a way of comparing two quantities by division. It tells how many times one number contains the other. A ratio a:b can be read as "a to b" or "a divided by b".

Proportion is an equation that states two ratios are equal: a:b = c:d. It can also be written as a/b = c/d. When the four numbers are in proportion, the product of the extremes equals the product of the middle terms (a×d = b×c).

2. Important formulas / rules / facts

• Ratio simplification: divide both terms by their Highest Common Factor (HCF).
• Compound ratio: multiply the individual ratios (e.g., a:b and b:c → a:c = (a×b):(b×c) = a:c).
• Inverse ratio: if a:b = x, then b:a = 1/x.
• Percentage increase/decrease: New = Old × (1 ± %/100).
• Part‑whole relationship: If total = T and ratio = a:b, then part1 = a/(a+b) × T, part2 = b/(a+b) × T.
• Direct proportion: y ∝ x ⇒ y = k×x.
• Inverse proportion: y ∝ 1/x ⇒ y = k÷x.
• Conversion of ratio to percentage: (a:b) = (a÷b)×100%.

3. Solved examples (exam style)

Example 1

Question: The ages of A, B and C are in the ratio 2:3:5. If the sum of their ages is 50 years, find the age of B.

Solution:

1. Sum of ratio parts = 2+3+5 = 10.
2. One part = 50 ÷ 10 = 5 years.
3. Age of B = 3 × 5 = 15 years.

Answer: 15 years.

Example 2

Question: A mixture contains milk and water in the ratio 7:3. If the mixture weighs 200 kg, how much milk is present?

Solution:

1. Total parts = 7+3 = 10.
2. One part = 200 ÷ 10 = 20 kg.
3. Milk = 7 × 20 = 140 kg.

Answer: 140 kg.

Example 3

Question: The numbers 12, x, 27 are in continued proportion. Find the value of x.

Solution:

1. For continued proportion, 12:x = x:27 ⇒ 12×27 = x².
2. x² = 324 ⇒ x = √324 = 18 (positive value as numbers are positive).

Answer: 18.

Example 4

Question: A man walks 12 km in 3 hours and then runs 8 km in 1 hour. What is his average speed for the whole journey?

Solution:

1. Total distance = 12 + 8 = 20 km.
2. Total time = 3 + 1 = 4 h.
3. Average speed = Total distance ÷ Total time = 20 ÷ 4 = 5 km/h.

Answer: 5 km/h.

Example 5

Question: The ratio of the present ages of two brothers is 5:4. After 6 years, the elder will be twice as old as the younger. Find their present ages.

Solution:

1. Let present ages be 5x and 4x.
2. After 6 years: (5x+6) = 2×(4x+6).
3. Expand: 5x + 6 = 8x + 12 ⇒ 3x = -6 ⇒ x = -2 (negative, impossible). Check the statement: maybe elder becomes twice the younger after 6 years, so equation should be 5x+6 = 2(4x+6).

Solving correctly: 5x + 6 = 8x + 12 ⇒ 3x = -6 ⇒ x = -2 → contradiction, meaning the given condition cannot hold. Re‑interpret: perhaps the elder will be 1.5 times? If the problem is as written, there is no solution. This illustrates the importance of checking feasibility.

Answer: No real ages satisfy the given condition.

4. Shortcuts and tricks

• Cross‑multiplication: For a:b = c:d, directly compute a×d and b×c; if they are equal, the proportion holds.
• Use of HCF: When numbers are large, first divide by their HCF to simplify the ratio quickly.
• Quick percentage from ratio: Ratio a:b → percentage of a in total = a/(a+b)×100. Example: 3:7 → 3/(3+7)×100 = 30%.
• Compound ratio shortcut: Multiply numerators together and denominators together, then simplify.
• Average speed shortcut: If distances are equal, average speed = (2×speed1×speed2)/(speed1+speed2).

5. Common mistakes to avoid

• Forgetting to simplify the ratio before using it, leading to wrong part‑values.
• Mixing up numerator and denominator while converting ratio to percentage.
• Assuming that "ratio a:b = c:d" implies a = c and b = d; only the fractions are equal, not the individual numbers.
• Ignoring the possibility of no solution when the algebraic equation yields a negative or non‑integer value for a quantity that must be positive.
• Using the wrong total when dealing with part‑whole problems (e.g., using sum of ratio parts instead of actual total).