1.

The sum of two numbers is 18 and the sum of their reciprocals is 1/4.Find the numbers.

Answer»

Let the numbers are a and b.

Given that,

The sum of these numbers is 18 and sum of their reciprocals is 1/4.

Hence, 

a + b = 18 … (1)

And,

\(\frac{1}{a}\) + \(\frac{1}{b}\) = \(\frac{1}{4}\)

⇒ \(\frac{a+b}{ab}\) = \(\frac{1}{4}\)

⇒ ab = 4(a + b) 

= 4 × 18 = 72.

(∵ a + b = 18.)

Now,

(a − b)2 = (a + b)2 − 4ab 

= 182 − 4 × 72 

= 324– 288 

= 36.

∴ a − b = \(\sqrt{36}\) = 6 … (2)

Now,

Adding equations (1) and (2), we get 

(a + b) + (a – b) 

= 18 + 6 = 24 

⇒ 2a = 24 

⇒ a = 12. 

Now,

Putting a = 12 in equation (1), we get 

12 + b = 18 

⇒ b = 18 − 12 = 6. 

Hence, 

The numbers are 6 and 12.


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