Class 10 Maths Chapter 6
Ratio and Proportion
Important Questions

Class 10 Mathematics Chapter 6, 'Ratio and Proportion'. Ratio and proportion are fundamental concepts in mathematics that play a crucial role in various mathematical and real-world applications. In class 10 ratio and proportion , you will encounter questions related to ratio and proportion, which are important to understand for both exams and practical problem-solving. The ratio and proportion questions class 10 ICSE are designed as a valuable tool to help students build confidence in their mathematical skills, alleviate doubts, and overcome difficulties they might encounter while studying this topic. students can strengthen their grasp of this fundamental mathematical concept, making it easier for them to excel in their class 10 Mathematics exams. So, dive into these questions and embark on a journey toward academic success!"

Introduction

Introduction to Ratio

A ratio in ratio and proportion class 10 ICSE is a way to compare two or more quantities or numbers. It is typically expressed as a fraction, and it can represent a part-to-part or part-to-whole comparison. The ratio of two quantities a and b is usually written as a:b or a/b.
For example:
The ratio of boys to girls in a classroom is 3:2.
The ratio of red balls to blue balls in a bag is 4:5.

Introduction to Proportion

Proportion refers to the equality of two ratios. In other words, if two ratios are equal, they are said to be in proportion.
For example:
2:3 and 4:6 are in proportion because they reduce to the same fraction (2/3).
5:7 and 15:21 are in proportion because they reduce to the same fraction (5/7).

What are Ratio and Proportion?

In Ratio and proportion class 10 ICSE, you will encounter the concepts of ratio and proportion, which are fundamental mathematical ideas used to compare quantities and establish relationships between them. Here's a more detailed explanation of both concepts:

1. Ratio - "Comparing Things":
Imagine you have a box of chocolates, and you want to tell someone how many are dark chocolate and how many are milk chocolate. This is where ratio comes into play. Ratio is like describing the chocolates in a way that's easy to understand. If you have 3 dark chocolates and 2 milk chocolates, you can say the ratio is 3 to 2 or 3:2. It's like saying, "For every 3 dark chocolates, there are 2 milk chocolates."
‍
2. Proportion - "Balancing Act":
Now, let's say you have two buckets, one with water and the other with sand. You want to keep the water-to-sand ratio the same when you transfer them to smaller containers. This is what proportion helps you with. It's like keeping things in balance. If you had 2 cups of water and 4 cups of sand in one bucket and 3 cups of water and 6 cups of sand in the other, they are in proportion because both have a 2:4 or 1:2 ratio of water to sand. It's like saying, "To keep the balance, we need 1 cup of water for every 2 cups of sand."

Class 10 Ratio and Proportion Important Questions and Answers

Q1. The compounded ratio of 2 : 3 and 5: 7 is:

Options

(a) 7 : 10
(b) 9 : 8
(c) 10 : 21
(d) 14 : 15

Ans. (d) 10 : 21

Explanation:
Compounded ratio of 2 : 3 and 5 : 7
= \(\frac{2}{3}\) × \(\frac{5}{7}\) = \(\frac{10}{21}\)

Q2. The compounded ratio of 8 : 3 and 5 : 7 is:

Options

(a) 7 : 10
(b) 9 : 8
(c) 40 : 21
(d) 14 : 15

Ans. (b) 40 : 21

Explanation:
Compounded ratio of 2 : 3 and 5 : 7
= \(\frac{8}{3}× \frac{5}{7}\) = \(\frac{40}{21}\)

Q3. If a, b, c, d are in continued proportion, prove that a : d = triplicate ratio of (a – b) : (b – c).

Explanation:
Given, a, b, c, d are in continued proportion.
∴ \(\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k\) (say)
⇒  c = kd, b = kc =k²d, a = kb = k³d
Now, L.H.S. = a : d = \(\frac{a}{b} = \frac{k^3d}{d} = k^3\)
R.H.S. = Triplicate ratio of (a – b) : (b – c)
= \(\frac{(a - b)^3}{(b - c)^3}\) = \(\left(\frac{k^3d - k^2d}{k^3d - kd}\right)^2\) = \(\left(\frac{k^2d(k-1)}{kd(k-d)}\right)\) = k³
Thus, L.H.S. = R.H.S. Hence Proved.

Q4. If b is the mean proportional between a and c, prove that \(\frac{a^2 - b^2 + c^2}{a^{-2} - b^{2} + c^{2}}\) = b⁴ .

Explanation:
Since, b is the mean proportion between a and c. 
‍So, b⁴=ac.
L.H.S. = \(\dfrac{a^2 - b^2 + c^2}{a^{-2} - b^{-2} + c^{-2}}\)
= \(\dfrac{a^2 - b^2 + c^2}{\dfrac{1}{a^2} - \dfrac{1}{b^2} + \dfrac{1}{c^2}}\)
= \(\dfrac{(a^2 - b^2 + c^2)}{\dfrac{b^2c^2 - a^2c^2 + a^2b^2}{a^2b^2c^2}}\)
= \(\dfrac{a^2b^2c^2(a^2 - b^2 + c^2)}{b^2c^{-2} - b^4 + a^2b^2}\)
= \(\dfrac{b^4×b^2(a^2 - b^2 + c^2)}{b^2(c^2 - b^2 + a^2)}\)
= b⁴
= R.H.S.

Q5. If a line passes through the point (1, 2) and cuts off positive intercepts on the X-axis and the Y-axis in the ratio 2 : 3, then find the equation of the line and its slope.

Explanation:
 Let the required line be AB cut off positive intercepts OA and OB on X-axis and Y-axis respectively.
∵ OA : OB = 2 : 3, let OA = 2a and OB = 3a.
∴ A = (2a, 0) and B = (0, 3a)
Equation of a line is given as
y - y₂ = \(\dfrac{y_2 - y_1}{x_2 - x_1}\)(x - x₁)
⇒ y - 0 = \(\dfrac{3a - 0}{0 - 2a}\)(x - 2a)
⇒ y = - \(\dfrac{3a}{2a}\)(x - 2a)
⇒ 2y = – 3(x – 2a)
        ⇒ 2y = – 3x + 6a
        ⇒ 3x + 2y – 6a = 0 ...(i)
        ∵ It passes through P(1, 2),
            Putting x = 1, y = 2, we get
              3 × 1 + 2 × 2 – 6a = 0
        ⇒ 7 – 6a = 0
        ⇒ 6a = 7
⇒                 a = \(\dfrac{7}{6}\)
∴ From equation (i),
                                       3x + 2y - 6 × \(\dfrac{7}{6}\) = 0
⇒ 3x + 2y – 7 = 0, which is the required equation of line
⇒           slope of AB = -\(\dfrac{\displaystyle{Coefficient\space of\space x}}{\displaystyle{Coefficient\space of\space y}}\)= - \(\dfrac{3}{2}\).

ICSE Class 10 Maths Chapter wise Important Questions

Conclusion

 "Ratio and Proportion in ICSE Class 10 math is all about comparing things (ratio) and keeping balance (proportion). It's like having a superpower to solve real-life problems in cooking, shapes, money, and more. These skills help you make smart decisions every day!" Mastering these ratio and proportion questions class 10 ICSE equips you with valuable problem-solving skills for everyday situations, making it a crucial part of your math education.

Frequently Asked Questions

Q1 : What is a ratio, and how is it represented in mathematics?

Ans: A ratio in class 10 ratio and proportion is a way to compare two or more quantities. It's typically represented as a:b or a/b, where 'a' and 'b' are numbers.

Q2: What is the difference between a ratio and a proportion?

Ans:  A ratio is a comparison of quantities, while a proportion is an equation that shows that two ratios are equal.

Q3 : How do you simplify a ratio?

Ans: To simplify a ratio, divide both the numerator and denominator by their greatest common factor (GCF).

Q4 : What are the practical applications of ratio and proportion in everyday life?

Ans: Ratio and proportion are used in cooking, construction, finance, and various other real-life situations.

Q5 : How do you set up and solve proportion problems?

Ans: Cross-multiplication is a common technique to set up and solve proportion problems.