Question

What number must be added to each of 5, 9, 17, 27 so that the new numbers are in proportion?
Or, the sum of two numbers is 18 and the sum of their reciprocals is $\frac{1}{4}$. Find the numbers.

Answer 1

Let the required number be x.

Then, according to proportionality rule, we get:
$\frac{5+x}{9+x}=\frac{17+x}{27+x}$
⇒ (5 + x)(27 + x) = (17 + x)(9 + x)
⇒ 135 + 5x + 27x + x² = 153 + 17x + 9x + x²
⇒ 135 + 32x = 153 + 26x
⇒ 32x - 26x = 153 - 135
⇒ 6x = 18
⇒ x = 3
Hence, the required number is 3.

OR
Let the required numbers be x and y.
Then, x + y = 18
and $\frac{1}{x}+\frac{1}{y}=\frac{1}{4}$
$\Rightarrow \frac{\left(x+y\right)}{xy}=\frac{1}{4}$
$$\begin{gathered} \Rightarrow \frac{18}{xy}=\frac{1}{4} \\ \Rightarrow xy=72 \end{gathered}$$
∴ $\left(x-y\right)=\sqrt{{\left(x+y\right)}^2-4xy}$
$$\begin{gathered} =\sqrt{{18}^2-4\times 72} \\ =\sqrt{324-288} \\ =\sqrt{36} \\ =\pm 6 \end{gathered}$$
∴ x + y = 18 .....(i)
and x - y = 6 ......(ii)
Or
∴ x + y = 18 .....(iii)
and x - y = - 6 ......(iv)
On solving (i) and (ii), we get: x = 12 and y = 6.
On solving (iii) and (iv), we get: x = 6 and y = 12.
Hence, the required numbers are 12 and 6.