Ratio and Proportion

Concept:For four numbers $a$, $b$, $c$, $d$ to be in proportion, the ratio $a:b$ must equal $c:d$, i.e., $\frac{a}{b} = \frac{c}{d}$.Explanation:Let the required number to be added be $x$. The new numbers become $6+x$, $14+x$, $18+x$, $38+x$. Set up the proportion: $\frac{6+x}{14+x} = \frac{18+x}{38+x}$.Cross-multiply: $(6+x)(38+x) = (14+x)(18+x)$.Expand both sides:LHS: $6 \cdot 38 + 6x + 38x + x^2 = 228 + 44x + x^2$.RHS: $14 \cdot 18 + 14x + 18x + x^2 = 252 + 32x + x^2$.Cancel $x^2$ from both sides: $228 + 44x = 252 + 32x$.Bring like terms together: $44x - 32x = 252 - 228$.Simplify: $12x = 24$. Divide by 12: $x = 2$.Thus, the least number to be added is $2$.Answer:$2$