Ratio and Proportion Chapter Notes | Mathematics Class 10 ICSE PDF Download

Introduction

Ever wondered how we compare quantities or share things fairly? The chapter on Ratio and Proportion is like a mathematical toolkit that helps us understand relationships between numbers and quantities in a simple yet powerful way. Whether it's dividing a pizza among friends, calculating speeds, or solving real-world problems like fare increases, this chapter equips you with the skills to compare, scale, and balance quantities effectively. From understanding the basics of ratios to exploring the magic of proportions and their properties, get ready to dive into a world where numbers work together in harmony!

• Ratio and proportion are fundamental concepts in mathematics, building on what was learned in Classes 8 and 9.
• This chapter explores these concepts in greater detail, focusing on their properties and practical applications.
• A ratio compares two quantities of the same kind, while a proportion establishes equality between two ratios.

Ratio

• A ratio compares two quantities of the same kind and unit by dividing one by the other.
• Formula: If a and b are quantities of the same kind and b ≠ 0, the ratio is a/b, written as a:b (read as "a is to b").
• Key points about ratios:
  • Ratios have no units and are expressed as a:b.
  • a is the first term (antecedent), b is the second term (consequent), and b cannot be zero.
  • Multiplying or dividing both terms by the same non-zero number does not change the ratio.
  • Ratios should be in their lowest terms (H.C.F. of terms is 1).
  • The order of terms matters; a:b ≠ b:a unless a = b.

Example 1: The ratio 3:7 is in its lowest terms because the H.C.F. of 3 and 7 is 1. However, 4:20 is not, as their H.C.F. is 4, so it simplifies to 1:5.
Example 2: If 2x + 3y : 3x + 5y = 18:29, find x:y.

• Step 1: Write the ratio as a fraction: (2x + 3y)/(3x + 5y) = 18/29.
• Step 2: Cross-multiply: 29(2x + 3y) = 18(3x + 5y).
• Step 3: Expand: 58x + 87y = 54x + 90y.
• Step 4: Simplify: 58x - 54x = 90y - 87y → 4x = 3y.
• Step 5: Solve for the ratio: x/y = 3/4, so x:y = 3:4.

Increase (or Decrease) in a Ratio

• When a quantity changes, the ratio of the original to the new quantity describes the increase or decrease.
• Formula: If a quantity changes from a to b, the ratio is a:b. The new quantity = (b/a) × original quantity.
• Increase example: Price increases from ₹20 to ₹24, ratio is 20:24 = 5:6.
• Decrease example: Price decreases from ₹24 to ₹20, ratio is 24:20 = 6:5.
• Example: If the fare of a journey increases in the ratio 5:7 and the new fare is ₹1,421, find the increase.
  • Step 1: Original fare : Increased fare = 5:7.
  • Step 2: Let original fare = x. Then, 7x = 5 × 1,421.
  • Step 3: Solve: x = (5 × 1,421)/7 = ₹1,015.
  • Step 4: Increase = 1,421 - 1,015 = ₹406.

Commensurable and Incommensurable Quantities

• Quantities are commensurable if their ratio can be expressed as a ratio of two integers.
• Quantities are incommensurable if their ratio cannot be expressed as a ratio of two integers.
• Example: The ratio between 2⅓ and 3½ is (7/3):(7/2) = 2:3 (commensurable). The ratio between √3 and 5 is √3:5 (incommensurable).

Composition of Ratios

Ratios can be combined to form new ratios using specific rules.

(i) Compound Ratio

• The ratio of the product of antecedents to the product of consequents of two or more ratios.
• Formula: For ratios a:b and c:d, compound ratio = (a × c):(b × d).
• Example: Find the compound ratio of 3a:2b, 2m:n, and 4x:3y.
  • Step 1: Multiply antecedents: 3a × 2m × 4x = 24amx.
  • Step 2: Multiply consequents: 2b × n × 3y = 6bny.
  • Step 3: Form ratio: 24amx:6bny = 4amx:bny.

(ii) Duplicate Ratio

• The compound ratio of two equal ratios.
• Formula: Duplicate ratio of a:b = a²:b².
• Example: Duplicate ratio of 2:3 = 2²:3² = 4:9.

(iii) Triplicate Ratio

• The compound ratio of three equal ratios.
• Formula: Triplicate ratio of a:b = a³:b³.
• Example: Triplicate ratio of 2:3 = 2³:3³ = 8:27.

(iv) Sub-duplicate Ratio

• The ratio of the square roots of the terms.
• Formula: Sub-duplicate ratio of a:b = √a:√b.
• Example: Sub-duplicate ratio of 9:16 = √9:√16 = 3:4.

(v) Sub-triplicate Ratio

• The ratio of the cube roots of the terms.
• Formula: Sub-triplicate ratio of a:b = ∛a:∛b.
• Example: Sub-triplicate ratio of 27:64 = ∛27:∛64 = 3:4.

(vi) Reciprocal Ratio

• The ratio obtained by swapping the antecedent and consequent.
• Formula: Reciprocal ratio of a:b = 1/a:1/b = b:a (where a, b ≠ 0).
• Example: Reciprocal ratio of 3:5 = 5:3.

Proportion

• Four non-zero quantities a, b, c, d are in proportion if a:b = c:d, written as a:b::c:d (read as "a is to b as c is to d").
• Formula: a:b = c:d ⇔ a/b = c/d ⇔ a × d = b × c (product of extremes = product of means).

Key points:

• a, b, c, d are terms: a (first), b (second), c (third), d (fourth).
• a and d are extremes; b and c are means.
• d is the fourth proportional.
• a and b must be of the same kind and unit; c and d must be of the same kind and unit.

Example 1: 5 kg: 15 kg = ₹75:₹225 is a proportion because 5/15 = 75/225 = 1/3.
Example 2: Find the fourth proportional to 3, 6, and 4.5.

• Step 1: Let the fourth proportional be x. Then, 3:6 = 4.5:x.
• Step 2: Write as fraction: 3/6 = 4.5/x.
• Step 3: Cross-multiply: 3x = 6 × 4.5.
• Step 4: Solve: x = (6 × 4.5)/3 = 9.

Continued Proportion

• Three non-zero quantities a, b, c of the same kind and unit are in continued proportion if a:b = b:c.
• Formula: a:b = b:c ⇔ a/b = b/c ⇔ b² = ac.
• b is the mean proportional between a and c; c is the third proportional to a and b.
• For quantities a, b, c, d, e, … in continued proportion, a/b = b/c = c/d = d/e = …

Example: Find the mean proportional between 6.25 and 0.16.

• Step 1: Let the mean proportional be x. Then, 6.25:x = x:0.16.
• Step 2: Write as a fraction: 6.25/x = x/0.16.
• Step 3: Cross-multiply: x² = 6.25 × 0.16.
• Step 4: Solve: x² = 1, so x = 1.

Some Important Properties of Proportion

If a:b = c:d, several other proportions can be derived using fraction properties.

1. Invertendo

• If a:b = c:d, then b:a = d:c.
• Formula: a/b = c/d ⇔ b/a = d/c.

Proof:

• Start with a/b = c/d.
• Take reciprocals: b/a = d/c.

Example: If 2:3 = 4:6, then 3:2 = 6:4 (since 3/2 = 6/4).

2. Alternendo

• If a:b = c:d, then a:c = b:d.
• Formula: a/b = c/d ⇔ a/c = b/d.

Proof:

• Start with a/b = c/d.
• Cross-multiply: a × d = b × c.
• Rearrange: a/c = b/d.

Example: If 2:3 = 4:6, then 2:4 = 3:6 (since 2/4 = 3/6 = 1/2).

3. Componendo

• If a:b = c:d, then a+b:b = c+d:d.
• Formula: a/b = c/d ⇔ (a+b)/b = (c+d)/d.

Proof:

• Start with a/b = c/d.
• Add 1 to both sides: a/b + 1 = c/d + 1.
• Simplify: (a+b)/b = (c+d)/d.

Example: If 2:3 = 4:6, then (2+3):3 = (4+6):6 → 5:3 = 10:6.

4. Dividendo

• If a:b = c:d, then a-b:b = c-d:d.
• Formula: a/b = c/d ⇔ (a-b)/b = (c-d)/d.

Proof:

• Start with a/b = c/d.
• Subtract 1 from both sides: a/b - 1 = c/d - 1.
• Simplify: (a-b)/b = (c-d)/d.

Example: If 2:3 = 4:6, then (2-3):3 = (4-6):6 → -1:3 = -2:6.

5. Componendo and Dividendo

• If a:b = c:d, then a+b:a-b = c+d:c-d.
• Formula: a/b = c/d ⇔ (a+b)/(a-b) = (c+d)/(c-d).

Proof:

• From componendo: (a+b)/b = (c+d)/d (Equation I).
• From dividendo: (a-b)/b = (c-d)/d (Equation II).
• Divide Equation I by Equation II: (a+b)/(a-b) = (c+d)/(c-d).

Example: If (8x+13y):(8x-13y) = 9:7, find x:y.

• Step 1: Apply componendo and dividendo: (8x+13y + 8x-13y)/(8x+13y - 8x+13y) = (9+7)/(9-7).
• Step 2: Simplify: 16x/26y = 16/2.
• Step 3: Solve: x/y = (16/2) × (26/16) = 13/1, so x:y = 13:1.

Alternative method:

• Start with (8x+13y)/(8x-13y) = 9/7.
• Cross-multiply: 72x - 117y = 56x + 91y.
• Simplify: 16x = 208y → x/y = 208/16 = 13/1, so x:y = 13:1.

Direct Applications

Proportions can be used to solve problems by applying properties like componendo and dividendo.

Example 1: If a:b = c:d, show that 3a+2b:3a-2b = 3c+2d:3c-2d.

• Step 1: Given a/b = c/d, let a/b = c/d = k, so a = bk, c = dk.
• Step 2: Compute (3a+2b)/(3a-2b) = (3bk+2b)/(3bk-2b) = (3k+2)/(3k-2).
• Step 3: Compute (3c+2d)/(3c-2d) = (3dk+2d)/(3dk-2d) = (3k+2)/(3k-2).
• Step 4: Since both are equal, 3a+2b:3a-2b = 3c+2d:3c-2d.

Example 2: If p = 4xy/(x+y), show that (p+2x)/(p-2x) + (p+2y)/(p-2y) = 2.

• Step 1: Given p = 4xy/(x+y), compute (p+2x)/(p-2x) = (4xy/(x+y) + 2x)/(4xy/(x+y) - 2x).
• Step 2: Simplify numerator: 4xy + 2x(x+y) = 2x(3y+x).
• Step 3: Simplify denominator: 4xy - 2x(x+y) = 2x(y-x).
• Step 4: So, (p+2x)/(p-2x) = (3y+x)/(y-x).
• Step 5: Similarly, (p+2y)/(p-2y) = (3x+y)/(x-y).
• Step 6: Add: [(3y+x)/(y-x)] + [(3x+y)/(x-y)] = [(3y+x)-(3x+y)]/(y-x) = 2.

Alternative method:

• Express p/2x = 2y/(x+y) and p/2y = 2x/(x+y).
• Apply componendo and dividendo to get (p+2x)/(p-2x) = (x+3y)/(y-x).
• Similarly, (p+2y)/(p-2y) = (3x+y)/(x-y).
• Add and simplify to get 2.