Math, asked by Itzraisingstar, 5 months ago

Sum of two natural numbers is 8 and the difference of their reciprocals is \bold{\frac{2}{15} } . Find the numbers??

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Answers

Answered by Bᴇʏᴏɴᴅᴇʀ
62

Answer:-

\red{\bigstar} The numbers are \large\leadsto\boxed{\rm\purple{3 \: and \: 5}}

Given:-

Sum of two natural numbers is 8.

Difference of their reciprocals is 2/15

To Find:-

The numbers

Solution:-

Let the numbers be 'x' and 'y'.

According to the question:-

\sf x + y = 8 \dashrightarrow\bf\red{[eqn.i]}

\sf \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{2}{15} \dashrightarrow\bf\red{[eqn.ii]}

Taking eqn[i]:-

\longrightarrow \sf x + y = 8

\longrightarrow \sf x = 8 - y

Substituting this value of x in eqn[ii]:-

\\ \longrightarrow \sf \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{2}{15} \\

\\ \longrightarrow \sf \dfrac{1}{(8-y)} - \dfrac{1}{y} = \dfrac{2}{15}

Taking LCM:-

\\ \longrightarrow \sf \dfrac{y - (8-y)}{y(8-y)} = \dfrac{2}{15} \\

\\ \longrightarrow \sf \dfrac{y - 8 + y}{8y-y^2} = \dfrac{2}{15}

\\ \longrightarrow \sf \dfrac{2y - 8}{8y-y^2} = \dfrac{2}{15}

Cross multiplying:-

\\ \longrightarrow \sf 15 \times(2y-8) = 2 \times (8y-y^2)

\\ \longrightarrow \sf 30y - 120 = 16y - 2y^2

\\ \longrightarrow \sf 30y - 16y - 120 = - 2y^2

\\ \longrightarrow \sf 2y^2 - 14y - 120

\\ \longrightarrow \sf y^2 - 7y - 60

Splitting the middle term:-

\\ \longrightarrow \sf y^2 - 5y + 12y - 60

\\ \longrightarrow \sf y(y - 5) + 12(y - 5)

\\ \longrightarrow \sf (y-5)(y+12)

Now,

\\ \bigstar \: \sf y - 5 = 0

\\ \longrightarrow \sf y = 5

\\ \bigstar \: \sf y + 12 = 0

\\ \longrightarrow \sf y = -12

Here, y = -12 is not possible.

Hence,

\pink{\bigstar} \bf\green{y = 5}

Substituting the value of y in eqn[i]:-

\\ \longrightarrow \sf x + y = 8

\\ \longrightarrow \sf x + 5 = 8

\\ \longrightarrow \sf x = 8 - 5

\pink{\bigstar} \bf\green{x = 3}

Therefore, the numbers are 3 and 5.


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mddilshad11ab: Perfect
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Anonymous: Great!
Answered by Anonymous
278

Answer:

Given :

  • Sum of two natural numbers is 8 .

  • the difference of their reciprocals is 2/15 .

To Find :

  • Find the numbers??

Solution :

Let the numbers be a and b

According to the Question :

a + b = 8

  • b = 8 - a .......( i )

1/b - 1/a = 2/15

  • a - b = (2/15) × ab .........( ii )

Adding (I) and (ii), we get :

2a = 8 + 2ab/15

30a = 120 + 2ab

Substitute the value of b ;

Therefore,

30a = 120 + 2a(8 - a)

30a = 120 + 16a - 2a²

30a - 16a + 2a² = 120

2a² + 14a - 120 = 0

a² + 7a - 60 = 0

  • [ dividing by 2 throughout ]

a² + 12a - 5a = 0

(a + 12)(a - 5) = 0

a = -12 or a = 5

As "a" is a natural number, a = -12 is inadmissible then a = 5

Using (i), we get b = 8 - a

Substitute all Values :

b = 8 - 5

b = 3

The required numbers are 5 and 3

Verification:

Sum of given numbers = 5 + 3 = 8

Difference of reciprocals = 1/3 - 1/5 = 2/15


prince5132: good.
mddilshad11ab: Nice¶
BrainlyPopularman: Awesome
Anonymous: Amazing!
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