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The sum of two numbers is 16. The sum of their reciprocals is $\frac{1}{3}$ . Find the numbers.

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10, 14

6, 9

5, 10

4, 12

answer is D.

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Detailed Solution

Given,
The sum of two numbers is 16and the sum of their reciprocals is

13

.
Consider the one number as

x

.
Therefore, another number is =

16 − x

Then according to question,

1 x + 1 (16 − x) = 13 ⇒ 16 − x + x x (16 − x) = 13 ⇒ 16 x (16 − x) = 13 ⇒ 16 × 3 = x (16 − x) ⇒ 48 = 16 x − x 2

⇒ x 2 − 16 x + 48 = 0 ⇒ x 2 − 12 x − 4 x + 48 = 0 ⇒ x (x − 12) − 4 (x − 12) = 0 ⇒ (x − 12) (x − 4) = 0 … .. (2)

Either,

(x − 12) = 0 ⇒ x = 12

Or,

(x − 4) = 0 ⇒ x = 4

Therefore, if

x = 12

then other number is,

(16 − 12) = 4

x = 4

then other number is,

(16 − 4) = 12

Since, in every case the final numbers are same, then the numbers are

4, 12

Thus, the two numbers are

4, 12

.
Hence, the correct option is 4.

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