# Problem of the Week Problem A and Solution Pumpkin Patch

## Problem

The mass of a standard carving pumpkin is approximately $$12$$ kg. Lavina plans to sell the pumpkins she has grown at the farmer’s market. The table she has to display the pumpkins can support $$224$$ kg. If the mass of each of her pumpkins is $$12$$ kg, what is the largest number of pumpkins that Lavina can put on her table?

## Solution

We can make a table to calculate the total mass of various quantities of pumpkins.

 Number of Pumpkins Total Mass (in kg) $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$24$$ $$36$$ $$48$$ $$60$$ $$72$$ $$84$$ $$96$$ $$108$$ $$120$$
 Number of Pumpkins Total Mass (in kg) $$11$$ $$12$$ $$13$$ $$14$$ $$15$$ $$16$$ $$17$$ $$18$$ $$19$$ $$132$$ $$144$$ $$156$$ $$168$$ $$180$$ $$192$$ $$204$$ $$216$$ $$228$$

The total mass of $$19$$ pumpkins, which is $$228$$ kg, exceeds the capacity of the table. So the largest number of pumpkins Lavina can fit safely on her table is $$18$$.

Having to make a table counting from $$1$$ to $$19$$ takes quite a bit of work. Alternatively, we could try to reduce the work by narrowing the search area. We can use easier numbers such as multiples of $$10$$ to find a narrower range to check. We see that $$10 \times 12 = 120$$ and $$20 \times 12 = 240$$. From this we know that the answer must be between $$10$$ and $$20$$ pumpkins. So instead of starting our table with $$1$$ pumpkin, we could start it with $$10$$ pumpkins.

We might also notice that the number we are looking for ($$224$$ kg) is much closer to $$240$$ than $$120$$. So rather than counting up, we could count down from $$240$$ in a table.

 Number of Pumpkins Total Mass (in kg) $$20$$ $$19$$ $$18$$ $$240$$ $$228$$ $$216$$

Again, from this result we can conclude that the largest number of pumpkins she can put on her table is $$18$$.

Teacher’s Notes

In this problem we said that the pumpkins all had the same mass of $$12$$ kg. However, in reality, we would not expect each of the pumpkins to have exactly the same mass. A more realistic statement would be that the mass of each pumpkin is approximately $$12$$ kg, but that makes the problem a bit trickier. If we had rounded to the nearest kilogram, that means the mass of each pumpkin could be greater than or equal to $$11.5$$ kg and less than $$12.5$$ kg.

Let’s assume that all the pumpkins have a mass of no more than $$11.6$$ kg, which is still approximately $$12$$ kg. In this case we see that $$11.6 \times 19 = 220.4$$ kg, so we could fit $$19$$ pumpkins on the table.

Let’s assume that all the pumpkins have a mass of no less than $$12.45$$ kg, which is still approximately $$12$$ kg. In this case we see that $$12.45 \times 18 = 224.1$$ kg, which is more than the capacity of the table.

Sometimes we need to recognize a margin of error to reflect that our physical world does not always fit into our nice and neat world of mathematical problems.