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There are two candles of same length and same size. Both of them burn at uniform rate. The first one burns in $5\,hr$ and the second one burns in $3\,h$. Both the candles are lit together. After how many minutes the length of the first candle is $3$ times that of the other?
$90$
$120$
$135$
$150$
Solution
(d)
We have, length and size of two candles are same. Let $L$ be the length of candles.
Given, first candle burns in $5 h$ and second candle burns in $3 h$.
In one hours length of candles are $\frac{L}{5}$ and $\frac{L}{3}$, respectively.
Let after time $t h$ the length of candles are $L_1$ and $L_2$
$\therefore \quad L_1=L-\frac{L}{5} t$ and $L_2=L-\frac{L}{3} t$
According to the problem,
$L_1=3 L_2$
$\therefore \quad L-\frac{L}{5} t=3\left(L-\frac{L}{3} t\right)$
$\Rightarrow 1-\frac{1}{5} t=3-t \Rightarrow t\left(1-\frac{1}{5}\right)=3-1$
$\Rightarrow \quad \frac{4 t}{5}=2 \Rightarrow t=\frac{5}{2} h$
$\Rightarrow \quad t=\frac{5}{2} \times 60=150\,min$
