Question

what least number must be added to each one of 6,14,18,38 to make them in proportional

Answer 1

let it be x.

6+x : 14+x :: 18+x : 38+x
⇒ (6+x)(38+x) = (14+x)(18+x)
⇒ 228 + 44x + x² = 252 + 32x + x²
⇒ 44x - 32x = 252 - 228
⇒ 12x = 24
⇒ x = 24/12 = 2

you should add 2 to each.

Answer 2

Answer:

The least number which should be added to make the numbers proportional is 2.

Solution:

Let the least no. that is to be added be x.

According to the question,

$\begin{array} { r } { ( x + 6 ) : ( x + 14 ) : ( x + 18 ) : ( x + 38 ) } \\\\ { \frac { x + 6 } { x + 14 } = \frac { x + 18 } { x + 38 } } \\\\ { ( x + 6 ) ( x + 38 ) = ( x + 14 ) ( x + 18 ) } \\\\ { x ^ { 2 } + 6 x + 38 x + 228 = x ^ { 2 } + 14 x + 18 x + 252 } \end{array}$

$\begin{array} { c } { \mathbb {  } x ^ { 2 } + 44 x + 228 = x ^ { 2 } + 32 x + 252 } \\\\ { x ^ { 2 } + 44 x - x ^ { 2 } - 32 x = 252 - 228 } \\\\ { 12 x = 24 } \\\\ { x = \frac { 24 } { 12 } = 2 } \end{array}$

Thus, the least no. that is to be added is = 2  

After adding the least number, the numbers will be proportional to each other.

The numbers which are in proportional are  

$\begin{aligned} ( x + 6 ) : ( x + 14 ) : : ( x + 18 ) : ( & ( x + 38 ) \\\\ ( 2 + 6 ) : ( 2 + 14 ) : : ( 2 + 18 ) & : ( 2 + 38 ) \\\\ 8 : 16 & : : 20 : 40 \end{aligned}$

Thus the numbers become proportional to each other after adding the number 2.