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The Difference of Two Natural Numbers is 5 and the Difference of Their Reciprocals is 1/10. Find the Numbers - Mathematics

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The difference of two natural numbers is 5 and the difference of their reciprocals is 1/10. Find the numbers

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Solution

Let the two natural numbers be x and y such that x > y.

Given:
Difference between the natural numbers = 5
∴x−y=5 ...(i)

Difference of their reciprocals = 110 (given)

`1/y−1/x=1/10`

`⇒(x−y)/(xy)=1/10`

`⇒5/(xy)=1/10`

⇒xy=50 ...(ii)

Putting the value of x from equation (i) in equation (ii), we get
(y+5)y=50

⇒y2+5y−50=0

⇒y2+10y−5y−50=0

⇒y(y+10)−5(y+10)=0

⇒(y−5)(y+10)=0

⇒y=5 or −10

As y is a natural number, therefore y = 5

Other natural number = y + 5 = 5 + 5 = 10

Thus, the two natural numbers are 5 and 10.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Substitution Method
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