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Higher Algebra Hall & Knight Proportion (Chapter 2) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 2 ‘Proportion’ have been formulated to include all the important topics related to Proportion for competitive exams like NEET and JEE. The chapter deals with the basics of Proportion like the comparison between the algebraic and geometrical definition of proportion, incommensurable quantities and continued proportional. The chapter of Higher Algebra by Hall & Knight contains comprehensive theoretical explanations and solved examples to help you gather a better understanding of Euclid’s definition of Proportion, mean proportional and third mean proportionality. 

Chapter 2 ‘Proportion’ of Higher Algebra by Hall & Knight includes 1 exercise which has 26 unsolved questions. This exercise contains questions related to direct and inverse proportion, many of which you would have dealt with in Class 11 and Class 12 Maths and Physics. For instance, finding the time taken by an object in free fall, ratio of thicknesses of two objects, and finding the relationship between two quantities on the basis of how their values change due to each other. Once you solve these kinds of questions, you will learn to tackle problems which deal with assignable quantities, propositions, componendo and dividendo which are often required for JEE preparation. 

Hall & Knight Algebra Mathematics book with solutions have been organized to ensure that you do not waste your time overlooking out for various resources and invest your efforts in performing better in your exams. Our subject experts have made sure that get a prolific, engaging and hassle-free learning experience. With these 100% accurate solutions from our dedicated team and your sincere efforts, no hurdle can stop you from improving your grades and acquiring your goals.

Important Topics for Hall & Knight Higher Algebra Solutions Chapter 2: Proportions

Proportional

In this chapter, you will be introduced to the concept of proportional. In case when the given two ratios are equal to each other, the four quantities which compose those ratios are known to be proportional. You would have got the basic idea of proportional in your junior classes. In this chapter, you will get an upgraded knowledge regarding them and concepts associated with them. to understand proportional we can take an example of a / b = c / d. Here a, b, c and d are proportional because the terms a and b or ratio one are equal to the term c and d of ratio two. This can be said as “a is to b as c is to d” and they all can be written as 

a : b : : c : d 

or 

a : b = c : d

here a and d are referred to as extremes or extreme values of the proportion and b and c are referred to as the means or the mean values of the proportion.

Finding the fourth proportional

In the chapter, once you are well versed with the idea of proportional, you will engage with the steps to find the fourth proportional in case if one ratio and one term of the other ratio is given. 

In case of given four proportions, the product of the mean values of that particular proportion is equal to the product of the extreme value of the same proportion, and this very concept is utilized in finding the fourth proportional of a series.

For example, if you are provided with four terms of proportions a, b, c and d such that b, c  and d are given, then, in this case, to find you will have to use the above-discussed method and therefore derive an as 

a = bd/c

Continued Proportion 

Taking an example, to begin with, 

In case of a : b : : b : c

We can notice that the first quantity is to second as the second quantity is to the third. This is known as continued proportion and it can also be expressed as 

a c = b2

Therefore in this chapter, we get to know that quantity are known to exist in continued proportion when the first quantity is to the second quantity as the second quantity is to the third quantity and so on.  

Here, you will also learn that the quantity b from the example is referred as the mean proportional between the quantities a and c and c is said to be the third proportional to the quantities a and b. This concept will further be utilized in many questions from the example exercise and has been demonstrated in the solved examples from the chapter too. 

Exercise-wise Discussion of  Higher Algebra by Hall and Knight Solutions Chapter 2: Proportions

  • This chapter of Hall and Knight Higher Algebra By H.S. Hall and S.R. Knight consists of just one exercise which is titled as “Example II” and has 26 unsolved questions. 
  • The exercise begins with a very basic question where you are required to find the fourth proportional of the given series.
  • There are questions related to the concept of mean proportion. 
  • The exercise also includes questions where you need to simplify the terms in order to prove they are equal and equate the left-hand side value to that given on the right-hand side. 
  • There are questions based on continuous proportions added to the exercise while some questions based on the theory of componendo and dividendo.
  • The last portion of the exercise has word problems which deal with the comparison of values, proportional derivations and ratio compositions. 

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