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# help

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Two candles of equal length are lit at the same time. One candle takes 6 hours to burn out, and the other takes 9 hours to burn out. After how much time will the slower burning candle be exactly twice as long as the faster burning one?

Jan 18, 2020

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Two candles of equal length are lit at the same time.

One candle takes 6 hours to burn out, and the other takes 9 hours to burn out.

After how much time will the slower burning candle be exactly twice as long as the faster burning one?

$$\begin{array}{|rcll|} \hline \text{Rest fast candle}&=& 1Candle - \dfrac{1candle}{6h}\times t \\ \text{Rest slow candle}&=& 1Candle - \dfrac{1candle}{9h}\times t \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 2\times \text{Rest fast candle } &=& \text{Rest slow candle} \\\\ 2\times \left( 1Candle - \dfrac{1candle}{6h}\times t\right) &=& 1Candle - \dfrac{1candle}{9h}\times t \\\\ 2 - \dfrac{2}{6}\times t &=& 1 - \dfrac{1}{9}\times t \\\\ 2 - \dfrac{1}{3}\times t &=& 1 - \dfrac{1}{9}\times t \\\\ \dfrac{1}{3}\times t- \dfrac{1}{9}\times t &=& 1 \\\\ \dfrac{(9-3)}{3*9}\times t &=& 1 \\\\ \dfrac{6}{3*9}\times t &=& 1 \\\\ \dfrac{2}{9}\times t &=& 1 \\\\ t &=& \dfrac{9}{2} \\\\ \mathbf{t} &=& \mathbf{4.5~ hours} \\ \hline \end{array}$$ Jan 19, 2020