Question

the difference of two no. is 9 and the difference of their reciprocal is 1/10 find the no. ​

Answer 1

Solution :

Let the two number be r and m.

A/q

$\longrightarrow\rm{r-m=9}\\\\\longrightarrow\rm{r=9+m..................(1)}$

&

$\longrightarrow\rm{\dfrac{1}{m} -\dfrac{1}{r} =\dfrac{1}{10} }\\\\\\\longrightarrow\rm{\dfrac{r-m}{rm} =\dfrac{1}{10} }\\\\\\\longrightarrow\rm{10(r-m)=rm}\\\\\\\longrightarrow\rm{10r-10m=rm}\\\\\\\longrightarrow\rm{10(9+m)-10m=(9+m)(m)\:\:[from(1)]}\\\\\\\longrightarrow\rm{90\cancel{+10m-10m}=9m+m^{2} }\\\\\\\longrightarrow\rm{90=9m+m^{2} }\\\\\\\longrightarrow\rm{m^{2} +9m-90=0\:\:\:[factorise]}\\\\\\\longrightarrow\rm{m^{2} +15m-6m-90=0}\\\\\\\longrightarrow\rm{m(m+15)-6(m+15)=0}$

$\longrightarrow\rm{(m+15)(m-6)=0}\\\\\\\longrightarrow\rm{m+15=0\:\:\:Or\:\:\:m-6=0}\\\\\\\longrightarrow\rm{\pink{m\neq -15\:\:\:Or\:\:\:m=6}}$

We know that negative value isn't acceptable.

Putting the value of m = 6 in equation (1),we get;

$\longrightarrow\rm{r=9+6}\\\\\longrightarrow\rm{\pink{r=15}}$

Thus;

The two number will be r = 15 and m = 6 .

Answer 2

GIVEN:

• Difference of two numbers is 9
• Difference of reciprocal of two numbers is ⅒

LET:

• Both the numbers be x & y

TO FIND:

• Both the numbers ( x , y )

SOLUTION:

Given,

x - y = 9

x = 9 + y……… ( i )

__________________

→ 1/x - 1/y = ⅒

→ y - x/xy = ⅒

→ 10y - 10x = xy

→ 10y - 10(9 + y) = (9 + y)y ( °.° Equation 1 )

→ 10y - 90 - 10y = 9y + y²

→ - 90 = 9y + y²

→ y² + 9y + 90 = 0

Now, we got a quadratic equation

→ y² + 15y - 6y +90 = 0

→ y(y + 15) - 6(y + 15) = 0

→ (y + 15)(y - 6) = 0

→ y = -15 , y = 6

Substituting y = 6 in equation 1

→ x = 9 + 6

→ x = 15

★ ANSWERS:

• x = 15
• y = 6