### Question 8

SSC-CGL 2018 June 4 Shift 2

If x is subtracted from each of 23, 39, 32 and 56, then the numbers obtained in this sequence are in proportion. What will be the mean proportional between (x + 4) and (3x+ 1)?

- 10
- 15
- 14
- 12

### Solution 1

Try guess an $\displaystyle x$ such that

$\displaystyle \frac{23 - x}{39 - x} = \frac{32 - x}{56- x}$

After some hit and trial we can discover that with x = 5,

$\displaystyle \frac{18}{34} \equiv \frac{27}{51}$

Hence, $\displaystyle \sqrt{(x + 4)(3x + 1)} $ $\displaystyle \equiv \sqrt{(5 + 4)(3\cdot5 + 1)}$= 12 answer!

### Solution 2

if we can't guess, we can quickly find x by the componendo trick as shown below.We have been given

$\displaystyle \frac{23 - x}{39 - x} = \frac{32 - x}{56- x}$

Use componendo dividendo to get

$\displaystyle \frac{23 - x}{(39 - x) - (23 - x)} = \frac{32 - x}{(56- x) - (32 - x)}$

$\displaystyle \implies \frac{23 - x}{16} = \frac{32 - x}{24}$

$\displaystyle \implies \frac{23 - x}{32 - x} = \frac{16}{24}$

Again componendo and dividendo,

$\displaystyle \implies \frac{23 - x}{(32 - x) - (23 - x)} = \frac{16}{24 - 16}$

$\displaystyle \implies \frac{23 - x}{9} = 2$

$\displaystyle \implies x = 5$

Hence, $\displaystyle \sqrt{(x + 4)(3x + 1)} = $ 12 answer!

### Solution 3

Cycle through the options. Check if any $\displaystyle x$ can be found that

$\displaystyle \sqrt{(x + 4)(3x + 1)} = 10, 15, 14 \text{ or } 13\text{?}$

By inspection, using x = 5 satisfies with

$\displaystyle \sqrt{(5 + 4)(3\cdot5 + 1)} = 12$

Hence (d) is the answer!

$ \begin{aligned} &\:\underline{Ans} \end{aligned} $

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