Class 10 Mathematics - Ch 5: Algebraic Fractions
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Class 10 Mathematics - Ch 5: Algebraic Fractions
Official curriculum roadmap, subject/topic distribution, negative marking rules, pacing guidelines, and solved sample questions.
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335 Solved Questions (MCQs, Short & Long Questions, Blanks, True/False).
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1 to 3 minutes per question depending on question type (MCQ, Short, Long).
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1 to 5 marks per question aligned with official board examination rubrics.
💡 Strategic Preparation & Exam Hall Guidelines
To maximize your score on Class 10 Mathematics - Ch 5: Algebraic Fractions, candidates are advised to follow a structured three-pass approach. In the First Pass, solve all direct recall and formula-based questions within 30 seconds each to secure foundational marks. In the Second Pass, tackle multi-step analytical and quantitative reasoning problems. In the Third Pass, review marked questions and verify calculations.
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Review the solved problems below to understand question phrasing, answer choices, and step-by-step solution logic prior to starting the full interactive practice drill:
Sample Question 1
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (15 a x^3 y^2) / (25 a^2 x y^6)
✓ Correct Answer: 3x^2 / (5ay^4)
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{15 a x^3 y^2}{25 a^2 x y^6}$$</p>
<p><strong>Step 1: Simplify numerical coefficients:</strong></p>
<p>$$\frac{15}{25} = \frac{3 \times 5}{5 \times 5} = \frac{3}{5}$$</p>
<p><strong>Step 2: Apply laws of exponents for like variable bases ($a, x, y$):</strong></p>
<p>$$\frac{a^1}{a^2} = \frac{1}{a^{2-1}} = \frac{1}{a}$$</p>
<p>$$\frac{x^3}{x^1} = x^{3-1} = x^2$$</p>
<p>$$\frac{y^2}{y^6} = \frac{1}{y^{6-2}} = \frac{1}{y^4}$$</p>
<p><strong>Step 3: Combine all simplified factors:</strong></p>
<p>$$\frac{3 \cdot x^2}{5 \cdot a \cdot y^4} = \frac{3x^2}{5ay^4}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{3x^2}{5ay^4}}$$</p>
<p>$$\frac{15 a x^3 y^2}{25 a^2 x y^6}$$</p>
<p><strong>Step 1: Simplify numerical coefficients:</strong></p>
<p>$$\frac{15}{25} = \frac{3 \times 5}{5 \times 5} = \frac{3}{5}$$</p>
<p><strong>Step 2: Apply laws of exponents for like variable bases ($a, x, y$):</strong></p>
<p>$$\frac{a^1}{a^2} = \frac{1}{a^{2-1}} = \frac{1}{a}$$</p>
<p>$$\frac{x^3}{x^1} = x^{3-1} = x^2$$</p>
<p>$$\frac{y^2}{y^6} = \frac{1}{y^{6-2}} = \frac{1}{y^4}$$</p>
<p><strong>Step 3: Combine all simplified factors:</strong></p>
<p>$$\frac{3 \cdot x^2}{5 \cdot a \cdot y^4} = \frac{3x^2}{5ay^4}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{3x^2}{5ay^4}}$$</p>
Sample Question 2
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (38 k^2 p^3 m^4) / (57 k^3 p m^2)
✓ Correct Answer: 2p^2 m^2 / (3k)
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{38 k^2 p^3 m^4}{57 k^3 p m^2}$$</p>
<p><strong>Step 1: Divide numerical coefficients by GCD(38, 57) = 19:</strong></p>
<p>$$\frac{38}{57} = \frac{2 \times 19}{3 \times 19} = \frac{2}{3}$$</p>
<p><strong>Step 2: Simplify algebraic powers:</strong></p>
<p>$$\frac{k^2}{k^3} = \frac{1}{k}, \quad \frac{p^3}{p^1} = p^2, \quad \frac{m^4}{m^2} = m^2$$</p>
<p><strong>Step 3: Combine factors:</strong></p>
<p>$$\frac{2 p^2 m^2}{3k}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{2p^2 m^2}{3k}}$$</p>
<p>$$\frac{38 k^2 p^3 m^4}{57 k^3 p m^2}$$</p>
<p><strong>Step 1: Divide numerical coefficients by GCD(38, 57) = 19:</strong></p>
<p>$$\frac{38}{57} = \frac{2 \times 19}{3 \times 19} = \frac{2}{3}$$</p>
<p><strong>Step 2: Simplify algebraic powers:</strong></p>
<p>$$\frac{k^2}{k^3} = \frac{1}{k}, \quad \frac{p^3}{p^1} = p^2, \quad \frac{m^4}{m^2} = m^2$$</p>
<p><strong>Step 3: Combine factors:</strong></p>
<p>$$\frac{2 p^2 m^2}{3k}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{2p^2 m^2}{3k}}$$</p>
Sample Question 3
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (46 l^3 m^4 n^5) / (69 l^2 m^3 n^4)
✓ Correct Answer: 2lmn / 3
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{46 l^3 m^4 n^5}{69 l^2 m^3 n^4}$$</p>
<p><strong>Step 1: Simplify numerical coefficients by dividing by GCD(46, 69) = 23:</strong></p>
<p>$$\frac{46}{69} = \frac{2 \times 23}{3 \times 23} = \frac{2}{3}$$</p>
<p><strong>Step 2: Simplify powers of variables $l, m, n$:</strong></p>
<p>$$\frac{l^3}{l^2} = l^{3-2} = l$$</p>
<p>$$\frac{m^4}{m^3} = m^{4-3} = m$$</p>
<p>$$\frac{n^5}{n^4} = n^{5-4} = n$$</p>
<p><strong>Step 3: Combine all terms:</strong></p>
<p>$$\frac{2 l m n}{3}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{2lmn}{3}}$$</p>
<p>$$\frac{46 l^3 m^4 n^5}{69 l^2 m^3 n^4}$$</p>
<p><strong>Step 1: Simplify numerical coefficients by dividing by GCD(46, 69) = 23:</strong></p>
<p>$$\frac{46}{69} = \frac{2 \times 23}{3 \times 23} = \frac{2}{3}$$</p>
<p><strong>Step 2: Simplify powers of variables $l, m, n$:</strong></p>
<p>$$\frac{l^3}{l^2} = l^{3-2} = l$$</p>
<p>$$\frac{m^4}{m^3} = m^{4-3} = m$$</p>
<p>$$\frac{n^5}{n^4} = n^{5-4} = n$$</p>
<p><strong>Step 3: Combine all terms:</strong></p>
<p>$$\frac{2 l m n}{3}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{2lmn}{3}}$$</p>
Sample Question 4
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (3 a b c) / (15 a^2 b^2 c)
✓ Correct Answer: 1 / (5ab)
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{3abc}{15a^2 b^2 c}$$</p>
<p><strong>Step 1: Divide numerical coefficients by 3:</strong></p>
<p>$$\frac{3}{15} = \frac{1}{5}$$</p>
<p><strong>Step 2: Cancel identical terms and subtract exponents:</strong></p>
<p>$$\frac{a}{a^2} = \frac{1}{a}, \quad \frac{b}{b^2} = \frac{1}{b}, \quad \frac{c}{c} = 1$$</p>
<p><strong>Step 3: Form the simplified fraction:</strong></p>
<p>$$\frac{1}{5ab}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{1}{5ab}}$$</p>
<p>$$\frac{3abc}{15a^2 b^2 c}$$</p>
<p><strong>Step 1: Divide numerical coefficients by 3:</strong></p>
<p>$$\frac{3}{15} = \frac{1}{5}$$</p>
<p><strong>Step 2: Cancel identical terms and subtract exponents:</strong></p>
<p>$$\frac{a}{a^2} = \frac{1}{a}, \quad \frac{b}{b^2} = \frac{1}{b}, \quad \frac{c}{c} = 1$$</p>
<p><strong>Step 3: Form the simplified fraction:</strong></p>
<p>$$\frac{1}{5ab}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{1}{5ab}}$$</p>
Sample Question 5
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (m n q) / (m p q)
✓ Correct Answer: n / p
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{mnq}{mpq}$$</p>
<p><strong>Step 1: Cancel common factors $m$ and $q$ in numerator and denominator:</strong></p>
<p>$$\frac{m \cdot n \cdot q}{m \cdot p \cdot q} = \frac{n}{p}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{n}{p}} \text{ (or } \mathbf{\frac{nq}{mp}} \text{)}$$</p>
<p>$$\frac{mnq}{mpq}$$</p>
<p><strong>Step 1: Cancel common factors $m$ and $q$ in numerator and denominator:</strong></p>
<p>$$\frac{m \cdot n \cdot q}{m \cdot p \cdot q} = \frac{n}{p}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{n}{p}} \text{ (or } \mathbf{\frac{nq}{mp}} \text{)}$$</p>
Sample Question 6
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (x - 3) / (3 - x)
✓ Correct Answer: -1
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{x - 3}{3 - x}$$</p>
<p><strong>Step 1: Factor out a negative sign ($-1$) from the denominator:</strong></p>
<p>$$3 - x = -(x - 3)$$</p>
<p><strong>Step 2: Rewrite the rational fraction:</strong></p>
<p>$$\frac{x - 3}{-(x - 3)}$$</p>
<p><strong>Step 3: Cancel $(x - 3)$ for $x \neq 3$:</strong></p>
<p>$$\frac{1}{-1} = -1$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{-1}$$</p>
<p>$$\frac{x - 3}{3 - x}$$</p>
<p><strong>Step 1: Factor out a negative sign ($-1$) from the denominator:</strong></p>
<p>$$3 - x = -(x - 3)$$</p>
<p><strong>Step 2: Rewrite the rational fraction:</strong></p>
<p>$$\frac{x - 3}{-(x - 3)}$$</p>
<p><strong>Step 3: Cancel $(x - 3)$ for $x \neq 3$:</strong></p>
<p>$$\frac{1}{-1} = -1$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{-1}$$</p>
Sample Question 7
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (x^2 - 81) / (x + 9)
✓ Correct Answer: x - 9
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{x^2 - 81}{x + 9}$$</p>
<p><strong>Step 1: Factorize numerator using difference of squares $a^2 - b^2 = (a-b)(a+b)$:</strong></p>
<p>$$x^2 - 81 = x^2 - 9^2 = (x - 9)(x + 9)$$</p>
<p><strong>Step 2: Substitute factored numerator:</strong></p>
<p>$$\frac{(x - 9)(x + 9)}{x + 9}$$</p>
<p><strong>Step 3: Cancel common binomial $(x + 9)$ (for $x \neq -9$):</strong></p>
<p>$$x - 9$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{x - 9}$$</p>
<p>$$\frac{x^2 - 81}{x + 9}$$</p>
<p><strong>Step 1: Factorize numerator using difference of squares $a^2 - b^2 = (a-b)(a+b)$:</strong></p>
<p>$$x^2 - 81 = x^2 - 9^2 = (x - 9)(x + 9)$$</p>
<p><strong>Step 2: Substitute factored numerator:</strong></p>
<p>$$\frac{(x - 9)(x + 9)}{x + 9}$$</p>
<p><strong>Step 3: Cancel common binomial $(x + 9)$ (for $x \neq -9$):</strong></p>
<p>$$x - 9$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{x - 9}$$</p>
Sample Question 8
Exercise 5.1 - Reduction of Algebraic Fractions to Lowest Form
MEDIUM • Short Question
Reduce the following rational expression to its lowest form: (x + 4) / (x^2 - 16)
✓ Correct Answer: 1 / (x - 4)
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Expression:</strong></p>
<p>$$\frac{x + 4}{x^2 - 16}$$</p>
<p><strong>Step 1: Factor the denominator using $a^2 - b^2 = (a-b)(a+b)$:</strong></p>
<p>$$x^2 - 16 = x^2 - 4^2 = (x - 4)(x + 4)$$</p>
<p><strong>Step 2: Rewrite fraction with factored denominator:</strong></p>
<p>$$\frac{x + 4}{(x - 4)(x + 4)}$$</p>
<p><strong>Step 3: Cancel $(x + 4)$ (for $x \neq -4$):</strong></p>
<p>$$\frac{1}{x - 4}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{1}{x - 4}}$$</p>
<p>$$\frac{x + 4}{x^2 - 16}$$</p>
<p><strong>Step 1: Factor the denominator using $a^2 - b^2 = (a-b)(a+b)$:</strong></p>
<p>$$x^2 - 16 = x^2 - 4^2 = (x - 4)(x + 4)$$</p>
<p><strong>Step 2: Rewrite fraction with factored denominator:</strong></p>
<p>$$\frac{x + 4}{(x - 4)(x + 4)}$$</p>
<p><strong>Step 3: Cancel $(x + 4)$ (for $x \neq -4$):</strong></p>
<p>$$\frac{1}{x - 4}$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{\frac{1}{x - 4}}$$</p>
Sample Question 9
Exercise 5.1 - Evaluation of Algebraic Expressions
MEDIUM • Short Question
Evaluate the expression (1/2) m v^2 when m = 18.75 and v = 5.6.
✓ Correct Answer: 294
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Formula:</strong></p>
<p>$$E_k = \frac{1}{2} m v^2$$</p>
<p><strong>Given Values:</strong> $m = 18.75$, $v = 5.6$.</p>
<p><strong>Step 1: Square the velocity $v$:</strong></p>
<p>$$v^2 = (5.6)^2 = 31.36$$</p>
<p><strong>Step 2: Multiply by $m$ and divide by $2$:</strong></p>
<p>$$\frac{1}{2} \times 18.75 \times 31.36 = 18.75 \times 15.68 = 294$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{294}$$</p>
<p>$$E_k = \frac{1}{2} m v^2$$</p>
<p><strong>Given Values:</strong> $m = 18.75$, $v = 5.6$.</p>
<p><strong>Step 1: Square the velocity $v$:</strong></p>
<p>$$v^2 = (5.6)^2 = 31.36$$</p>
<p><strong>Step 2: Multiply by $m$ and divide by $2$:</strong></p>
<p>$$\frac{1}{2} \times 18.75 \times 31.36 = 18.75 \times 15.68 = 294$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{294}$$</p>
Sample Question 10
Exercise 5.1 - Evaluation of Algebraic Expressions
MEDIUM • Short Question
Evaluate (4/3) pi r^3 when r = 2.1 and pi = 22/7.
✓ Correct Answer: 38.808
📖 Step-by-Step Solution & Conceptual Rationale:
<p><strong>Given Formula (Sphere Volume):</strong></p>
<p>$$V = \frac{4}{3} \pi r^3$$</p>
<p><strong>Given Values:</strong> $r = 2.1 = \frac{21}{10}$, $\pi = \frac{22}{7}$.</p>
<p><strong>Step 1: Compute $r^3$:</strong></p>
<p>$$r^3 = (2.1)^3 = 9.261 = \frac{9261}{1000}$$</p>
<p><strong>Step 2: Substitute and simplify:</strong></p>
<p>$$V = \frac{4}{3} \times \frac{22}{7} \times \frac{9261}{1000} = \frac{88 \times 441}{1000} = \frac{38808}{1000} = 38.808$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{38.808 \text{ or } 38\frac{101}{125}}$$</p>
<p>$$V = \frac{4}{3} \pi r^3$$</p>
<p><strong>Given Values:</strong> $r = 2.1 = \frac{21}{10}$, $\pi = \frac{22}{7}$.</p>
<p><strong>Step 1: Compute $r^3$:</strong></p>
<p>$$r^3 = (2.1)^3 = 9.261 = \frac{9261}{1000}$$</p>
<p><strong>Step 2: Substitute and simplify:</strong></p>
<p>$$V = \frac{4}{3} \times \frac{22}{7} \times \frac{9261}{1000} = \frac{88 \times 441}{1000} = \frac{38808}{1000} = 38.808$$</p>
<p><strong>Final Answer:</strong></p>
<p>$$\mathbf{38.808 \text{ or } 38\frac{101}{125}}$$</p>
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