# Problem of the Week Problem A and Solution Feeding the Pets

## Problem

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.

## Solution

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.