Question

the sum of reciprocal of two consecutive odd natural numbers is 12/35. find the numbers?​

Answer 1

Answer:

5and7

Step-by-step explanation:

let the no. be x, x+2

so1/x+1/x+2=12/35

(2x+2)35=12x(x+2)

35x+35=6x^2+12x

6x^2-23x-35=0

6x^2-30x+7x-35=0

6x(x-5)+7(x-5)=0

(6x+7)(x-5)=0

x=5 is natural no.

Answer 2

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

• The sum of reciprocal of two consecutive odd natural number is 12/35.

$\large\underline{\sf{To\:Find - }}$

• The two consecutive odd natural numbers.

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

We know that,

• Two consecutive odd natural numbers differ by 2,

So,

$\begin{gathered}\begin{gathered}\sf \: Let \: consecutive \: odd \: natural \: numbers \: be \: \begin{cases} &\sf{x} \\ &\sf{x + 2} \end{cases}\end{gathered}\end{gathered}$

According to statement,

Sum of the reciprocals of two consecutive odd natural numbers is 12/35.

$\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{x + 2} = \dfrac{12}{35}$

$\rm :\longmapsto\:\dfrac{x + 2 + x}{x(x + 2)} = \dfrac{12}{35}$

$\rm :\longmapsto\:\dfrac{2x + 2}{ {x}^{2} + 2x} = \dfrac{12}{35}$

$\rm :\longmapsto\: {12x}^{2} + 24x = 70x + 70$

$\rm :\longmapsto\:12 {x}^{2} - 46x - 70 = 0$

$\rm :\longmapsto\:2( {6x}^{2} - 23x - 35) = 0$

$\rm :\longmapsto\: {6x}^{2} - 23x - 35 = 0$

$\rm :\longmapsto\: {6x}^{2} - 30x + 7x - 35 = 0$

$\rm :\implies\:6x(x - 5) + 7(x - 5) = 0$

$\rm :\longmapsto\:(x - 5)(6x + 7) = 0$

$\bf\implies \:x = 5 \: \: \: or \: x = - \dfrac{7}{6} \: (rejected \: as \: x \cancel\in \: N)$

$\begin{gathered}\begin{gathered}\sf \: Hence, \: consecutive \: odd \: natural \: numbers - \begin{cases} &\sf{x = 5} \\ &\sf{x + 2 = 7} \end{cases}\end{gathered}\end{gathered}$

Additional Information :-

Writing System of Equations from Word Problems

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.