The sum of two numbers is 16 and the sum of their reciprocals is 1/3 . Find the numbers.
The sum of two numbers is 16 and the sum of their reciprocals is 1/3 . Find the numbers.
4 views
1 Answers
Let the larger number be x and the smaller number be y.
Then, we have:
x + y = 16 ……(i)
And, 1/x + 1/y = 1/3 ……(ii)
⇒3(x + y) = xy
⇒3 × 16 = xy [Since from (i), we have: x + y = 16]
∴ xy = 48 …….(iii)
We know:
(x – y)2 = (x + y)2 – 4xy
(x – y)2 = (16)2 – 4 × 48 = 256 – 192 = 64
∴ (x – y) = ±√64 = ±8
Since x is larger and y is smaller, we have:
x – y = 8 ………(iv)
On adding (i) and (iv), we get:
2x = 24
⇒x = 12
On substituting x = 12 in (i), we get:
12 + y = 16
⇒ y = (16 – 12) = 4
Hence, the required numbers are 12 and 4
4 views
Answered