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Math Notes For Class 10 Chapter 3 Ex 3.1 Solved & Quiz
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Math Notes For Class 10 Chapter 3 Ex 3.1 Solved & Quiz
In Matric Mathematics (Class 10), Chapter 3 focuses on the theory of Variations. Exercise 3.1 introduces the foundational concept of a Ratio and a Proportion, which are fundamental tools in algebra, calculus, and practical mathematics.
| Concept / Relation | Mathematical Notation | Key Formula / Identity |
|---|---|---|
| Ratio Definition | $a : b$ or $\frac{a}{b}$ | Requires $b \neq 0$ and same units |
| Equality of Ratios | $a : b = c : d$ | $\frac{a}{b} = \frac{c}{d}$ |
| Fundamental Property | $a : b :: c : d$ | $\text{Product of Extremes} = \text{Product of Means} \implies a \cdot d = b \cdot c$ |
| Homogeneous Ratio Evaluation | $\frac{xa + yb}{za + wb}$ | Divide numerator and denominator by $b$ to use $\frac{a}{b}$ |
Question 1: Express the following as a ratio $a : b$ and as a fraction in its simplest form $\frac{a}{b}$:
(i) $\text{Rs. } 750, \text{Rs. } 1250$
(ii) $450\text{ cm}, 3\text{ m}$
(iii) $4\text{ kg}, 2\text{ kg } 500\text{ g}$
Solution:
Part (i): Both quantities are already in the same unit (Rupees).
$$\text{Ratio} = 750 : 1250$$ $$\text{Fraction Form} = \frac{750}{1250}$$ Dividing numerator and denominator by $250$: $$\frac{750 \div 250}{1250 \div 250} = \frac{3}{5}$$Answer (i): $$\text{Ratio} = 3 : 5, \quad \text{Fraction} = \frac{3}{5}$$
Part (ii): Convert both quantities to centimeters ($1\text{ m} = 100\text{ cm}$).
$$3\text{ m} = 3 \times 100\text{ cm} = 300\text{ cm}$$ $$\text{Ratio} = 450 : 300$$ $$\text{Fraction Form} = \frac{450}{300}$$ Dividing numerator and denominator by $150$: $$\frac{450 \div 150}{300 \div 150} = \frac{3}{2}$$Answer (ii): $$\text{Ratio} = 3 : 2, \quad \text{Fraction} = \frac{3}{2}$$
Part (iii): Convert both quantities to grams ($1\text{ kg} = 1000\text{ g}$).
$$4\text{ kg} = 4 \times 1000\text{ g} = 4000\text{ g}$$ $$2\text{ kg } 500\text{ g} = (2 \times 1000) + 500 = 2500\text{ g}$$ $$\text{Ratio} = 4000 : 2500$$ $$\text{Fraction Form} = \frac{4000}{2500}$$ Dividing numerator and denominator by $500$: $$\frac{4000 \div 500}{2500 \div 500} = \frac{8}{5}$$Answer (iii): $$\text{Ratio} = 8 : 5, \quad \text{Fraction} = \frac{8}{5}$$
Question 2: In a class of $60$ students, $25$ are girls and the rest are boys. Find the ratio of:
(i) Boys to total students
(ii) Boys to girls
Solution:
Total students $= 60$
Number of girls $= 25$
Number of boys $= 60 - 25 = 35$
Part (i): Ratio of boys to total students:
$$\text{Ratio} = 35 : 60 = \frac{35}{60}$$ Dividing numerator and denominator by $5$: $$\frac{35 \div 5}{60 \div 5} = \frac{7}{12}$$Answer (i): $$\text{Ratio} = 7 : 12, \quad \text{Fraction} = \frac{7}{12}$$
Part (ii): Ratio of boys to girls:
$$\text{Ratio} = 35 : 25 = \frac{35}{25}$$ Dividing numerator and denominator by $5$: $$\frac{35 \div 5}{25 \div 5} = \frac{7}{5}$$Answer (ii): $$\text{Ratio} = 7 : 5, \quad \text{Fraction} = \frac{7}{5}$$
Question 3: Find the value of $x$ if $3(4x - 5) = 2(3x + 7)$.
Solution:
Given linear algebraic relation:
$$3(4x - 5) = 2(3x + 7)$$ Expand both sides using distributive property: $$12x - 15 = 6x + 14$$ Rearrange terms with $x$ on the left-hand side and constants on the right-hand side: $$12x - 6x = 14 + 15$$ $$6x = 29$$ Divide both sides by $6$: $$x = \frac{29}{6}$$Answer: $$x = \frac{29}{6}$$
Question 4: Find the value of $p$ if the ratios $2p + 5 : 3p + 4$ and $3 : 4$ are equal.
Solution:
Setting the two ratios equal to form a proportion:
$$(2p + 5) : (3p + 4) = 3 : 4$$ Express in fraction form: $$\frac{2p + 5}{3p + 4} = \frac{3}{4}$$ Cross-multiply: $$4(2p + 5) = 3(3p + 4)$$ Expand both sides: $$8p + 20 = 9p + 12$$ Rearrange terms: $$20 - 12 = 9p - 8p$$ $$p = 8$$Answer: $$p = 8$$
Question 5: If $a : b = 3 : 4$, find the value of $(4a + 5b) : (5a - 2b)$.
Solution:
Given $a : b = 3 : 4$, which means $\frac{a}{b} = \frac{3}{4}$.
We need to evaluate the expression:
$$\text{Value} = \frac{4a + 5b}{5a - 2b}$$ Divide numerator and denominator by $b$: $$\text{Value} = \frac{\frac{4a + 5b}{b}}{\frac{5a - 2b}{b}} = \frac{4\left(\frac{a}{b}\right) + 5}{5\left(\frac{a}{b}\right) - 2}$$ Substitute $\frac{a}{b} = \frac{3}{4}$ into the expression: $$\text{Value} = \frac{4\left(\frac{3}{4}\right) + 5}{5\left(\frac{3}{4}\right) - 2}$$ Simplify numerator and denominator separately: $$\text{Numerator} = 3 + 5 = 8$$ $$\text{Denominator} = \frac{15}{4} - 2 = \frac{15 - 8}{4} = \frac{7}{4}$$ Now combine numerator and denominator: $$\text{Value} = \frac{8}{\frac{7}{4}} = 8 \times \frac{4}{7} = \frac{32}{7}$$Answer: $$32 : 7 \quad \left(\text{or } \frac{32}{7}\right)$$
Question 6: If $x : y = 2 : 3$, evaluate the algebraic fraction $\frac{3x + 2y}{5x - y}$.
Solution:
Given $x : y = 2 : 3 \implies \frac{x}{y} = \frac{2}{3}$.
We need to find:
$$\frac{3x + 2y}{5x - y}$$ Divide numerator and denominator by $y$: $$\frac{\frac{3x + 2y}{y}}{\frac{5x - y}{y}} = \frac{3\left(\frac{x}{y}\right) + 2}{5\left(\frac{x}{y}\right) - 1}$$ Substitute $\frac{x}{y} = \frac{2}{3}$: $$\text{Numerator} = 3\left(\frac{2}{3}\right) + 2 = 2 + 2 = 4$$ $$\text{Denominator} = 5\left(\frac{2}{3}\right) - 1 = \frac{10}{3} - 1 = \frac{10 - 3}{3} = \frac{7}{3}$$ Combine numerator and denominator: $$\text{Value} = \frac{4}{\frac{7}{3}} = 4 \times \frac{3}{7} = \frac{12}{7}$$Answer: $$\frac{12}{7}$$
Q1: In the ratio $x : y$, the term $x$ is technically known as the:
Q2: What is the simplified form of the ratio $150\text{ g} : 1.5\text{ kg}$?
Q3: If $x : 4 = 9 : 12$, what is the value of $x$?
Q4: In the proportion $a : b = c : d$, the product $a \cdot d$ represents the:
Q5: What is the ratio of $15\text{ minutes}$ to $1\text{ hour}$ in simplest form?