Problem A and Solution

Feeding the Pets

I give my dog \(162\) g of dog food each meal. He eats two meals per day.

I give my cat \(95\) g of cat food each meal. She eats one meal per day.

The dog food comes in a \(5\) kg bag. The cat food comes in a \(2\) kg bag.

If I start a new bag of food for each pet on the same day, which bag will run out first? Justify your answer.

One way to determine which bag will run out first is to make a table that keeps track of how much food each pet consumes.

Since the dog eats two meals of \(162\) g each day, then he consumes \(162 + 162 = 324\) g per day. Since \(1\) kg = \(1000\) g, then \(2\) kg = \(2 \times 1000 = 2000\) g and \(5\) kg = \(5 \times 1000 = 5000\) g. We will continue tracking the amount of food consumed until we reach one of these limits.

Day |
Total Consumed by Cat (in
g) |
Total Consumed by Dog (in
g) |
---|---|---|

\(1\) | \(95\) | \(324\) |

\(2\) | \(190\) | \(648\) |

\(3\) | \(285\) | \(972\) |

\(4\) | \(380\) | \(1296\) |

\(5\) | \(475\) | \(1620\) |

\(6\) | \(570\) | \(1944\) |

\(7\) | \(665\) | \(2268\) |

\(8\) | \(760\) | \(2592\) |

\(9\) | \(855\) | \(2916\) |

\(10\) | \(950\) | \(3240\) |

\(11\) | \(1045\) | \(3564\) |

\(12\) | \(1140\) | \(3888\) |

\(13\) | \(1235\) | \(4212\) |

\(14\) | \(1330\) | \(4536\) |

\(15\) | \(1425\) | \(4860\) |

\(16\) | \(1520\) | \(5184\) |

On Day \(16\) the dog will have consumed more than \(5000\) g of food, but the cat has not yet consumed \(2000\) g of food. So the dog food bag will run out first.

Alternatively, we could use estimation to determine which bag will
run out first. Since \(95\) g is less
than \(100\) g, then the cat food will
last at least \(20\) days since \(20 \times 100 = 2000\). Since \(324\) g is greater than \(250\) g, then the dog food will last less
than \(20\) days since \(20 \times 250 = 5000\). Therefore, the cat
food will last longer than the dog food, so the dog food bag will run
out first.