Ratio, Proportion & Variation

Ratio and proportion drive partnership, mixture and speed problems in CDS. The trick is to convert a ratio into parts (a common multiplier k) and work with whole numbers.

Ratios and proportion

• A ratio a:b can be written as ak : bk; solve by finding k.
• Proportion a:b :: c:d means a/b = c/d, i.e. ad = bc (product of extremes = product of means).
• Mean proportional between a and b = sqrt(ab); third proportional to a, b = b^2/a.

Variation

• Direct variation: y = kx (y increases as x increases).
• Inverse variation: y = k/x (y decreases as x increases); xy = constant.

Applications

• Partnership: profit is shared in the ratio of (capital x time) for each partner.
• Componendo-dividendo: if a/b = c/d then (a+b)/(a-b) = (c+d)/(c-d) — speeds up messy proportion sums.

Exam Tricks & Tips

• 🎯 Turn every ratio into 'parts' with a multiplier k, solve for k, then read off the actual values.
• 🎯 In proportion, cross-multiply: product of extremes = product of means (ad = bc).
• 🎯 Mean proportional = sqrt(ab); third proportional = b^2/a — memorise both.
• 🎯 Partnership profit splits as capital x time; equalise the time factor if partners join at different months.
• 🎯 Use componendo-dividendo to jump from a/b to (a+b)/(a-b) in one move.
• ❌ Common mistake: adding quantities from different ratios directly without first making a common term equal (e.g. a:b and b:c need the shared b matched before combining to a:b:c).

Expected exam pattern

Divide-in-a-ratio, proportion, partnership and direct/inverse variation questions; 6-10 marks and fast with the parts method.

Quick recap

Use the multiplier-k parts method, cross-multiply proportions, recall mean/third proportional formulae, split partnership by capital x time, and match the common term before combining ratios.

Cover the answer, recall, then check. 11 cards on ratio and proportion.

Q1. What does a : b :: c : d imply?
A1. a/b = c/d, i.e. ad = bc (product of extremes = product of means).

Q2. Mean proportional between a and b?
A2. sqrt(ab).

Q3. Third proportional to a and b?
A3. b^2/a.

Q4. Divide 600 in the ratio 2:3. What are the parts?
A4. 5 parts -> 120 each; 2:3 = 240 and 360.

Q5. Equation form of inverse variation?
A5. y = k/x, i.e. xy = constant.

Q6. In partnership, in what ratio is profit shared?
A6. In the ratio of (capital x time) for each partner.

Q7. Componendo-dividendo: from a/b = c/d, what follows?
A7. (a+b)/(a-b) = (c+d)/(c-d).

Q8. If a:b = 2:3 and b:c = 4:5, find a:b:c.
A8. Match b: a:b = 8:12, b:c = 12:15, so a:b:c = 8:12:15.

Q9. Direct variation equation?
A9. y = kx.

Q10. Fourth proportional to a, b, c?
A10. (b x c)/a.

Q11. A and B invest 5000 for 12 months and 6000 for 10 months. Profit ratio?
A11. 5000x12 : 6000x10 = 60000 : 60000 = 1:1.

Ratio and proportion express how quantities compare and scale together. They are central to CDS arithmetic and underpin partnership, mixtures, and speed problems, so fluency here pays across many topics.

Core idea / what this tests

This tests manipulation of ratios (a : b), the properties of proportions (a : b :: c : d), and direct/inverse variation. The key skill is scaling quantities by a common factor and using the "product of means = product of extremes" rule.

Deep explanation

Beginner — ratios and their scaling

A ratio a : b compares two quantities; it can be scaled by any common factor, so 2 : 3 = 4 : 6 = 200 : 300. When a total is divided in a ratio a : b, the parts are [a/(a+b)] and [b/(a+b)] of the total. Introduce a variable k: if incomes are in ratio 3 : 5, write them as 3k and 5k, then use the given condition to find k.

Intermediate — proportion properties

In a proportion a : b :: c : d (i.e., a/b = c/d):

• Product of extremes = product of means: a × d = b × c (cross-multiplication).
• Mean proportional between a and b is √(ab); the third proportional to a and b is b²/a; the fourth proportional to a, b, c is bc/a.
• Compound ratio of a : b and c : d is (ac) : (bd).

Advanced — direct and inverse variation

• Direct variation: y ∝ x means y = kx; if one doubles, so does the other (more workers → more output).
• Inverse variation: y ∝ 1/x means xy = k; if one doubles, the other halves (more workers → less time).
  Combining ratios: to merge a : b and b : c into a : b : c, scale so the common term b matches (LCM of the two b-values). This is essential for three-way partnership and mixture problems.

Worked example

Two numbers are in the ratio 3 : 5. If 4 is added to each, the ratio becomes 5 : 7. Find the numbers.

Step 1 — let the numbers be 3k and 5k (the variable-k method).
Step 2 — apply the new condition: (3k + 4)/(5k + 4) = 5/7.
Step 3 — cross-multiply: 7(3k + 4) = 5(5k + 4) → 21k + 28 = 25k + 20.
Step 4 — solve: 28 − 20 = 25k − 21k → 8 = 4k → k = 2.
Step 5 — the numbers are 3k = 6 and 5k = 10. Check: (6+4):(10+4) = 10:14 = 5:7 ✓.
Answer: 6 and 10. The k-substitution converted a ratio problem into a linear equation.

CDS relevance

Ratio-proportion is a high-frequency CDS topic and the language of partnership, mixtures, and rate problems. The variable-k method and cross-multiplication make these questions systematic, and recognising direct vs inverse variation avoids setup errors in many word problems.

Exam tricks & shortcuts

• Introduce k: turn ratio parts into ak, bk and use the given condition.
• Cross-multiply proportions: extremes' product = means' product.
• Mean proportional = √(ab); third proportional = b²/a; fourth = bc/a.
• Direct variation (y = kx) vs inverse (xy = k) — decide which before computing.
• To combine a : b and b : c, scale on the common term.

Key formulas

• a:b = c:d ⇒ ad = bc (product of extremes = product of means).
• Continued proportion a:b:c ⇒ b² = ac; mean proportional = √(ac).
• If a/b = c/d, then (a+b)/b = (c+d)/d (componendo), (a−b)/(a+b) = (c−d)/(c+d).
• Dividing X in ratio a:b: parts = Xa/(a+b), Xb/(a+b).
• Direct proportion y ∝ x ⇒ y = kx; inverse y ∝ 1/x ⇒ xy = k.
• Combine ratios a:b and b:c to a:b:c via the common term.

Usage: cross-multiply proportions; use componendo–dividendo for (a±b) forms.

Worked Example

Problem: Divide ₹1200 among A, B and C in the ratio 2 : 3 : 5. How much does each receive?

Solution:
Step 1 — Add the ratio parts to find the total number of parts:
2 + 3 + 5 = 10 parts.

Step 2 — Find the value of one part:
one part = total amount / total parts = 1200 / 10 = ₹120.

Step 3 — Multiply each ratio number by the value of one part:
A = 2 × 120 = ₹240
B = 3 × 120 = ₹360
C = 5 × 120 = ₹600

Step 4 — Check: 240 + 360 + 600 = ₹1200 ✓, which matches the total.

Answer: A gets ₹240, B gets ₹360, and C gets ₹600.