You have two candles of equal height. The first one takes 6 hours to burn out while the other one takes 9 hours to burn out. One night before going to sleep, you lit both candles. Upon waking up, you noticed that one of the candles is now half the height of the other. How many minutes were you asleep for?

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Suppose,

The height of each candle = h

After each hour the first candle lose h/6 of it’s height

And the second candle lose h/9 of it’s height

Again suppose,

After x hours the first candle will be the half height of the other.

After x hours,

The height of the first candle will be = h - \cfrac{hx}{6}

The height of the second candle will be = h - \cfrac{hx}{9}

According to question,

2 \bigg( h - \cfrac{hx}{6}\bigg) = h - \cfrac{hx}{9}

=> 2h - \cfrac{hx}{3}= h - \cfrac{hx}{9}

=> h= \cfrac{hx}{3} - \cfrac{hx}{9}

=> h= \cfrac{2hx}{9}

=> x= \cfrac{9}{2} = 4.5

Therefore, I was asleep for 4.5 hours = 270 minutes \fbox{Ans}