If one small rabbit can dig a warren in nine hours, a medium rabbit can dig a warren in 6 hours, and a large rabbit can dig a warren in four and a half hours.

How long will it take one small, one medium and one large rabbit to dig a warren if they all work together?

Puzzle Copyright © Kevin Stone

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Hint

How much of a warren each rabbit dig in one hour?

Answer

2 hours.

Reasoning

The small rabbit can dig a warren in nine hours, so digs ^{1}/_{9} in one hour.

The medium rabbit can dig a warren in 6 hours, so digs ^{1}/_{6} in one hour.

The large rabbit can dig a warren in four and a half hours, so digs ^{1}/_{4.5} in one hour.

1 Hour = ^{1}/_{9} + ^{1}/_{6} + ^{1}/_{4.5}

To make the denominators all the same, we can multiply through by ^{18}/_{18} to give:

1 Hour = ^{2}/_{18} + ^{3}/_{18} + ^{4}/_{18}

1 Hour = ^{9}/_{18}

1 Hour = ^{1}/_{2}

So in one hour, they have dug half a warren, the whole warren will therefore take 2 hours.

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