the sum of numbers is 11 and sum of their reciprocals is 11/28.find the numbers
Answers
Answer:
The sum of two numbers is 11 and the sum of their reciprocal is 11/28.
Let the numbers be x and y respectively.
Sum of numbers is 11.
⇒ x + y = 11.... (i)
⇒ y = 11 - x .... (ii)
Sum of reciprocals is 11/28.
⇒ \dfrac{1}{\text{x}} + \dfrac{1}{\text{y}}= \dfrac{11}{28}
x
1
+
y
1
=
28
11
.... (iii)
Now, on solving (iii),
\begin{gathered}\dfrac{1}{\text{x}} + \dfrac{1}{\text{y}}= \dfrac{11}{28} \\ \\ \frac{ \text {x + y }}{ \text{xy}} = \frac{11}{28} \\ \\ \bf on \: cross \: multiplying : \\ \\28( \text{x + y) = 11xy}\end{gathered}
x
1
+
y
1
=
28
11
xy
x + y
=
28
11
oncrossmultiplying:
28(x + y) = 11xy
Putting the value of (i) and (ii) here, we get -
\begin{gathered}28 * 11 = 11 \text{x(11 - x)} \\ \\ \implies 28 = \text{x(11 - x)} \\ \\ \implies 28 = 11 \text{x - x}{}^{2} \\ \\ \implies \text{x} {}^{2} - 11\text x + 28 \\ \\ \implies \text{x} {}^{2} - 7\text{x - 4x} + 28 = 0 \\ \\ \implies \text{x(x - 7) - 4(x - 7) = 0 } \\ \\ \implies \text{(x - 7)(x - 4) = 0} \\ \\ \boxed{\therefore \bf x = 7 \: or \: x = 4}\end{gathered}
28∗11=11x(11 - x)
⟹28=x(11 - x)
⟹28=11x - x
2
⟹x
2
−11x+28
⟹x
2
−7x - 4x+28=0
⟹x(x - 7) - 4(x - 7) = 0
⟹(x - 7)(x - 4) = 0
∴x=7orx=4
Hence, the required numbers are 7 and 4.