Notes of Class 10 math

Unit 3 Variations Comprehensive Solutions

In Unit 3 Variations, students encounter foundational concepts in mathematics, such as ratios, proportions, and types of variation, which are vital for both advanced studies and real-world applications. This chapter covers various theorems like Invertendo, Alternendo, Componendo, Dividendo, and Componendo & Dividendo, offering systematic methods to solve proportion-based problems. In this post, we provide detailed solutions to each exercise in Unit 3 Variations, guiding students through these complex concepts step-by-step. With these solutions, students can better understand the logic behind each topic, reinforcing their skills in ratio and proportion to handle challenging problems confidently.

Exercise 3.1, Exercise 3.2

Multiple Choice Questions Test for Unit 3

To help solidify your understanding of Unit 3 Variations, we’ve included a multiple-choice quiz that covers the main topics, including ratios, proportions, variations, and key theorems. This test is designed to reinforce the concepts covered in this unit and help assess your mastery of each key idea.

Unit 3 MCQs Test for 10 class math

1 / 15

In a ratio a:b, a is called

Relation

Antecedent

Consequent

None of these

2 / 15

In a ratio $x : y$ , $y$ is called:

relation

antecedent

consequent

None of these

3 / 15

In a proportion $a : b :: c : d$ , $a$ and $d$ are called

means

extremes

third proportional

None of these

4 / 15

In a proportion $a : b :: c : d$ , $b$ and $c$ are called:

means

extremes

fourth proportional

None of these

5 / 15

In continued proportion $a : b = b : c$ , $ac = b^2$ , $b$ is said to be ________ proportional between $a$ and $c$ .

third

fourth

means

none of these

6 / 15

In continued proportion $a : b = b : c$ , $c$ is said to be ________ proportional to $a$ and $b$ .

third

fourth

means

none of these

7 / 15

Find $x$ in proportion $4 : x :: 5 : 15$ :

75/4

4/3

3/4

8 / 15

If $u \propto v^2$ , then:

$$ u=v^2 $$

$$ u=kv^2 $$

$$ uv^2=k $$

$$ uv^2=1 $$

9 / 15

If $y^2 \propto \frac{1}{x^3}$ , then:

$$ 𝑦^2 = 𝑘/ 𝑥^3 $$

$$ y ^2 = 1/x^3 $$

$$ 𝑦^2 = 𝑥^2 $$

$$ 𝑦^2 = 𝑘 𝑥^3 $$

10 / 15

If $\frac{u}{v} = \frac{v}{w} = k$ then:

$$ 𝑢 = 𝑤 𝑘^2 $$

$$ 𝑢 = 𝑣 𝑘^2 $$

$$ 𝑢 = w^2 𝑘 $$

$$ 𝑢 = 𝑣^2 𝑘 $$

11 / 15

The third proportional of $x^2$ and $y^2$ is:

$$ 𝑦^2 /𝑥^2 $$

$$ 𝑥^2 𝑦^2 $$

$$ 𝑦^4/ 𝑥^2 $$

$$ 𝑦^2 /𝑥^4 $$

12 / 15

The fourth proportional $w$ of $x : y :: v : w$ is:

$$ 𝑥𝑦 /𝑣 $$

$$ 𝑣𝑦 /𝑥 $$

$$ xyv $$

$$ 𝑥/𝑦𝑣 $$

13 / 15

If $a : b = x : y$ , then alternando property is:

$$ a/x = b/y $$

$$ a/b = x/y $$

$$(a+b)/b = (x+y)/y $$

$$ (a−b)/x = (x−y)/y $$

14 / 15

If $a : b = x : y$ , then invertendo property is:

$$ 𝑎 / 𝑥 = 𝑏 / 𝑦 $$

$$ (𝑎 − 𝑏)/ 𝑏 = (𝑥 − 𝑦)/𝑦 $$

$$ (𝑎 + 𝑏)/ 𝑏 = (𝑥 + 𝑦)/𝑦 $$

$$ 𝑏/𝑎 = 𝑦/𝑥 $$

15 / 15

If $\frac{a}{b} = \frac{c}{d}$ , then componendo property is:

$$ (𝑎 + 𝑏)/ 𝑎 = (𝑐 + 𝑑)/ 𝑑 $$

$$ (𝑎 - 𝑏)/ 𝑏 = (𝑐 - 𝑑)/ 𝑑 $$

$$ ad/bc $$

$$ (𝑎 - 𝑏)/ 𝑎 = (𝑐 - 𝑑)/ 𝑑 $$

Your score is

The average score is 71%

Things to learn in Unit 3 Variations

Definitions:

Ratio: A comparison of two quantities by division.
A ratio is a mathematical concept used to compare two quantities by dividing one quantity by another. It expresses how many times one quantity is contained within the other. Ratios can be written in different forms, such as fractions $ \frac{a}{b} \text{ with a colon } a:b $, or using the word “to” (e.g., “a to b”). Ratios are used to represent relationships between quantities in various contexts, such as speed (distance/time), density (mass/volume), and mixing ingredients in a recipe. The ratio helps simplify the relationship between two values, making it easier to understand proportions and compare their relative sizes.
Proportion: An equation that states two ratios are equal.
A proportion is a mathematical statement that shows two ratios are equal to each other. In other words, it expresses the idea that two quantities are in the same relative relationship. Proportions are often written in the form $\frac{a}{b} = \frac{c}{d}$, where the two ratios $\frac{a}{b}$ and $\frac{c}{d}$ are equal. Proportions can be solved by cross-multiplying, which involves multiplying the numerator of one ratio by the denominator of the other ratio. Proportions are used in solving problems related to scaling, map reading, mixtures, and finding unknown quantities in real-life situations. The concept of proportion is fundamental in understanding direct variation and solving practical problems involving comparisons.
Variation: The relationship between quantities, which can be direct or inverse.
Variation refers to how one quantity changes in relation to another. In direct variation, when one quantity increases, the other also increases in a proportional manner, and this relationship is represented by the equation $y=kx$, where $k$ is the constant of proportionality. In inverse variation, when one quantity increases, the other decreases, and the relationship is given by the equation$y= \frac{k}{x}$, where the product of the two quantities remains constant. Variation helps model many real-world situations, such as calculating speed or scaling.

unit 3 variations definitions of Proportion.

Proportion Types:

3rd, 4th, Mean, and Continued Proportion: Techniques to find and work with proportions.

Theorems:

Invertendo: If \[\frac{a}{b} = \frac{c}{d}, then \frac{b}{a} = \frac{d}{c}\].
Alternendo: If \[\frac{a}{b} = \frac{c}{d}, then \frac{a}{c} = \frac{b}{d}\].
Componendo: If \[\frac{a}{b} = \frac{c}{d}, then \frac{a+b}{b} = \frac{c+d}{d}\].
Dividendo: If \[\frac{a}{b} = \frac{c}{d}, then \frac{a-b}{b} = \frac{c-d}{d}\].
Componendo & Dividendo: If \[\frac{a}{b} = \frac{c}{d}, then \frac{a+b}{a-b} = \frac{c+d}{c-d}.\]

Joint Variation: Definition and solving problems related to the relationship where a quantity varies directly or inversely with multiple variables.

K-method: Using the constant of proportionality (k) to prove conditional equalities in proportions.

Real-life Applications: Solving practical problems using the concepts of ratio, proportion, and variation.

Here are the important formulas used in this unit 3 variations. If you want to learn more please proceed to the solutions of unit 4.

Unit 3 Variations builds essential skills in understanding mathematical relationships that are widely applicable in advanced math and real-life scenarios. By grasping the concepts of ratio, proportion, and theorems, students will be well-prepared for more challenging topics in algebra and physics. This chapter also lays the groundwork for solving diverse problems in various fields, from finance to engineering. For further practice, check our Class 10 math exercises for more resources, or continue exploring Solutions of unit 5 to strengthen your understanding. For a deeper dive into practical applications, you may find this article on Cuemath useful.

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Multiple Choice Questions Test for Unit 3

Things to learn in Unit 3 Variations

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