103rd Problem 2016

$\begin{aligned} \dfrac{8}{4} & = \dfrac{6}{y +1}\\ \\ \dfrac{2}{1} & = \dfrac{6}{y + 1}\\ \\ 2(y + 1) & = 6\\ 2y + 2 & = 6\\ 2y & = 6 - 2\\ 2y & = 4\\ y & = \dfrac{4}{2}\\ \\ y & = \boxed{2} \end{aligned}$

Easiest way:

The denominator of 6 should be half of it. The half of 6 is 3.

y + 1 = 3
y = 2

$\begin{matrix} \frac{8}{4}=\frac{6}{y+1}&&\\\\ 8\left(y+1\right)=4\cdot 6&&\frac{a}{b}=\frac{c}{d}\rightarrow ad=bc\\ 8y+8=24&&a(b+c)=ab+ac\\ 8y+8-8=24-8&&\\ 8y=16&\\ \frac{8y}{8}=\frac{16}{8}&&\\ \mathbf{y=2}&&\\ &Check:&\\ &\frac{8}{4}=\frac{6}{y+1}&\\\\ &\frac{8}{4}=\frac{6}{(2)+1}&\\ &\frac{8}{4}=\frac{6}{3}&\\ &\frac{8\div 4}{4\div 4}=\frac{6\div 3}{3\div 3}&\\ &\frac{1}{2}=\frac{1}{2}&\\ &\textbf{True}&\\ \end{matrix}$