The denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator and denominator respectively. The ratio of numerator to denominator is 3:5 find the fraction.
ANSWER
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Hint:We will start our solution by first defining the variables x and y and then use the formula of fraction and rationalisation. We will consider x as numerator and y as denominator and then by applying the condition given in the question, we get y = 2x - 1 and then solve accordingly.
Complete step-by-step answer: A fraction is a part of a whole. It is also a ratio between two integers, two integers are separated by a solidus (/) or a vinculum (_). A fraction has two parts. The number on the top of the line is called the numerator. The number below the line is called the denominator. Numerator tells how many equal parts of the whole or collection are taken. Denominator shows the total divisible number of equal parts the whole into or the total number of equal parts which are there in a collection. Let the numerator be x and denominator be y. Given that denominator of fraction is 1 less than twice its numerator. Hence, we get, \[ \Rightarrow y{\text{ }} = {\text{ }}2x{\text{ }} - {\text{ }}1\,\,\,\,\,\, \ldots ..{\text{ }}\left( 1 \right)\] Also given that if 1 is added to numerator and denominator respectively the ratio of numerator to denominator is 3:5. Hence, we get, \[ \Rightarrow x + 1:y + 1 = 3:5\,\,\,\,\,\, \ldots ..{\text{ }}\left( 2 \right)\] Substituting equation (1) in equation (2), we get, $ \Rightarrow x + 1:\left( {2x - 1} \right) - 1 = 3:5$ $ \Rightarrow x + 1:2x = 3:5\,\,\,\,\,\,\, \ldots ..{\text{ (3)}}$ Converting equation (3) in fraction form, we get, $ \Rightarrow \dfrac{{x + 1}}{{2x}} = \dfrac{3}{5}$ Using cross-multiplication concept, we get, $\begin{gathered} \Rightarrow 5(x + 1) = 3(2x) \\ \Rightarrow 5x + 5 = 6x \\ \Rightarrow 6x - 5x = 5 \\ \end{gathered} $ $ \Rightarrow x = 5\,\,\,\,\,\, \ldots ..{\text{ }}\left( 4 \right)$ Thus, x = 5. Substituting value of ‘x’ from equation (4) in equation (1), we get value of ‘y’: $\begin{gathered} \Rightarrow y = 2x - 1 \\ \Rightarrow y = 2(5) - 1 \\ \Rightarrow y = 9 \\ \end{gathered} $ Thus, y = 9. Therefore, the value of the numerator is 5 and the denominator is 9. Which can be represented as $\dfrac{x}{y} = \dfrac{5}{9}$ Hence, the fraction is $\dfrac{5}{9}$.
Note: For solving this type of questions, knowing the concept of fraction is the key here. For solving this type of questions, one should be aware of the basic concept of ratio, numerator and denominator. In this question we can directly replace x and y with x and 2x – 1 as given in question.