Two candles of equal height but …

Let height of each candle = $x\ unit.$
First candle burns off in 6 hours.
Second candle burns off in 8 hours.
Height of 1st candle after burning for 1 hr = $\frac{x}{6}$$unit$
and height of 2nd candle after burning for 1 hr = $\frac{x}{8}$$unit$
Let the required time $= y\ hrs$.
Length of 1st candle burnt after y hrs = $\frac{y \times x}{6}$$unit$
Height of 1st candle left = $\left(x-\frac{x y}{6}\right)$
Length of 2nd candle burnt after y hrs = $\left(\frac{y \times x}{8}\right)$$unit$
Height of 2nd candle left = $\left(x-\frac{x y}{8}\right)$
According to the question,
Height of 1st candle = $\frac{1}{2} \times$Height of 2nd candle
$\Rightarrow \quad x-\frac{x y}{6}=\frac{1}{2}\left(x-\frac{x y}{8}\right)$
$\Rightarrow \quad x\left(1-\frac{y}{6}\right)=\frac{1}{2} x\left(1-\frac{y}{8}\right)$
$1-\frac{y}{6}=\frac{1}{2}\left(1-\frac{y}{8}\right)$
$\Rightarrow \quad 2-\frac{y}{3}=1-\frac{y}{8}$
$2 -1 =$ $\frac{y}{3}-\frac{y}{8}$
1 = $\frac{8 y-3 y}{24}$
$\Rightarrow$ $24 = 5y$
$\Rightarrow$ $y =$ $\frac{24}{5}$
$y = 4.8 hours = 4\ hours\ 48\ minutes.$