Question

Sum of two rational numbers is 8 and difference of their reciprocal is 2/15. Find the numbers.​

Answer 1

$\large\underline{\sf{Solution-}}$

↝ Let assume that

• First rational number be x

and

• Second rational number be y

↝ Further, assume that x > y.

According to statement,

↝ Sum of two rational numbers is 8.

$\rm :\longmapsto\:x + y = 8$

$\bf :\longmapsto\: y = 8 - x - - - (1)$

Note :-

$\: \: \: \: \: \: \: \: \: \: \pink{ \boxed{ \bf \: If \: x > y \: then \: \dfrac{1}{y} > \dfrac{1}{x} }}$

According to second condition,

↝ Difference of their reciprocals is 2/15.

$\rm :\longmapsto\:\dfrac{1}{y} - \dfrac{1}{x} = \dfrac{2}{15}$

$\rm :\longmapsto\:\dfrac{1}{8 - x} - \dfrac{1}{x} = \dfrac{2}{15} \: \: \: \: \: \: \: \{ \: using \: (1) \: \}$

$\rm :\longmapsto\:\dfrac{x - (8 - x)}{x(8 - x)} = \dfrac{2}{15}$

$\rm :\longmapsto\:\dfrac{x - 8 + x}{x(8 - x)} = \dfrac{2}{15}$

$\rm :\longmapsto\:\dfrac{2x -8}{x(8 - x)} = \dfrac{2}{15}$

$\rm :\longmapsto\:\dfrac{2(x -4)}{x(8 - x)} = \dfrac{2}{15}$

$\rm :\longmapsto\:\dfrac{x -4}{x(8 - x)} = \dfrac{1}{15}$

$\rm :\longmapsto\:15x - 60 = 8x - {x}^{2}$

$\rm :\longmapsto\: {x}^{2} + 7x - 60 = 0$

$\rm :\longmapsto\: {x}^{2} + 12x - 5x - 60 = 0$

$\rm :\longmapsto\:x(x + 12) - 5(x + 12) = 0$

$\rm :\longmapsto\:(x + 12)(x - 5) = 0$

$\rm :\implies\:x = - 12 \: \: \: or \: \: \: x = 5$

So, corresponding values of y is

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y = 8 - x \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 5 & \sf 3 \\ \\ \sf - 12 & \sf 20 \end{array}} \\ \end{gathered}$

Hence,

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:Numbers \: are \: \bf\: -\begin{cases} &\sf{5 \: and \: 3} \\ &\sf{or}\\ &\sf{ - 12 \: and \: 20} \end{cases}\end{gathered}\end{gathered}$

Basic Concept Used :-

Writing Systems of  Equation from Word Problem.

1. Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find.

2. Translate the problem to an equation.

• Assign a variable (or variables) to represent the unknown.

• Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Answer 2

Step-by-step explanation:

⛄⛄⛄Geological evidence indicates that dinosaurs became extinct at the boundary between the Cretaceous and Paleogene eras, about 66 million years ago, at a time when there was worldwide environmental change resulting from the impact of a large celestial object with the Earth and/or from vast volcanic eruptions.

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