Question

the sum of two numbers is 20 and the sum of their reciprocals is 5/24 find the numbers

Answer 1

let one no.be x then the other will be 20-x
since the sum of their reciprocal are 5/24 then
1/x+1/20-x=5/24
solving we get a quadratic equation as x^-20x+96=0
factoring we get (x+8)(x-12)=0
SO THE NUMBERS ARE -8 and 28 OR 12 AND 8..

Answer 2

Answer:

$\large{\underline{\boxed{\sf 12 \: and \: 8}}}$

Step-by-step explanation:

Given that -

The sum of two numbers is 20 and the sum of their reciprocal is 5/24

Let the numbers be x and y respectively.

Sum of numbers is 20

⇒ x + y = 20.... (i)

⇒ y = 20 - x .... (ii)

Sum of reciprocals is 5/24.

⇒ $\dfrac{1}{\text{x}} + \dfrac{1}{\text{y}}= \dfrac{5}{24}$.... (iii)

Now, on solving (iii),

$\dfrac{1}{\text{x}} + \dfrac{1}{\text{y}}= \dfrac{5}{24} \\ \\ \frac{ \text {x + y }}{ \text{xy}} = \frac{5}{24} \\ \\ \bf on \: cross \: multiplying : \\ \\24( \text{x + y) = 5xy}$

Putting the value of (i) and (ii) here, we get -

$24 * 20 = 5 \text{x(20 - x)} \\ \\ \implies 96 = \text{x(20 - x)} \\ \\ \implies 96 = 20 \text{x - x}{}^{2} \\ \\ \implies \text{x} {}^{2} - 20\text x + 96 \\ \\ \implies \text{x} {}^{2} - 12\text{x - 8x} + 96 = 0 \\ \\ \implies \text{x(x - 12) - 8(x - 12) = 0 } \\ \\ \implies \text{(x - 12)(x - 8) = 0} \\ \\ \boxed{\therefore \bf x = 12 \: or \: x = 8}$

Hence, the required numbers are 12 and 8.