Question

The sum of two numbers is 18. The sum of their reciprocals is $\frac{1}{4}$. Find the numbers.

Answer 1

SOLUTION :

Given : Sum of two numbers is 18 & Sum of their reciprocals is ¼ .

Let the one number be x and the other number be (18 - x).

Their reciprocals be = 1/x & 1/(18 - x)

A.T.Q

1/x + 1/(18 - x) = 1/4

(18 - x + x)/(18 - x)x = ¼

[By taking L. C. M]

18/(18 - x)x = ¼

18/ (18x - x²) = 1/4

18 × 4 = 18x - x²

72 = 18x - x²

x² - 18x + 72 = 0

x² - 12x - 6x - 72 = 0

[By middle term splitting]

x(x - 12) - 6(x - 12) = 0

(x - 6) (x - 12) = 0

(x - 6)  = 0  or (x - 12) = 0

x = 6 or x = 12

Hence, the two numbers are 6 and 12.

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Answer 2

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

Let the one number be x and the other number be (18 - x).

Their reciprocals be = 1/x & 1/(18 - x)

A.T.Q

1/x + 1/(18 - x) = 1/4

(18 - x + x)/(18 - x)x = 1/4

[By taking L. C. M]

18/(18 - x)x = 1/4

18/ (18x - x²) = 1/4

18 × 4 = 18x - x²

72 = 18x - x²

x² - 18x + 72 = 0

x² - 12x - 6x - 72 = 0

[By middle term splitting]

x(x - 12) - 6(x - 12) = 0

(x - 6) (x - 12) = 0

(x - 6)  = 0  or (x - 12) = 0

x = 6 or x = 12

Hence the two numbers are 6 and 12.