Pumpkin Pie Problem
2 out of 3
Peter wanted to make two pies for a party. His mother had a recipe for him to use. However, she always made 80 pies at a time. Did Peter have enough ingredients to make two pumpkin pies?
Problem
Pumpkin Pie Problem printable sheet
Peter, the pumpkin eater, wanted to make two pies for a party. His mother, a professional pie maker, had a recipe for him to use. However, she always made 80 pies at a time. She used:
10 dozen eggs
27 litres of condensed milk
480 tablespoons of sugar
100 teaspoons of cinnamon
140 cups of pumpkin
Peter looked in the cupboard and found:
4 cups of pumpkin
2 eggs
1 $\frac{1}{2}$ teaspoons of cinnamon
$\frac{2}{3}$ of a litre of condensed milk
15 tablespoons of sugar
Did Peter have enough ingredients to make two pumpkin pies for the party or did he need to buy more?
Getting Started
How will you find out what you need to make two pumpkin pies?
What would you need for ten pies?
Student Solutions
It looks like Peter is going to have to go shopping!
A way to think about this problem was to find out what proportion of the original recipe Peter required to make the two pies. You might have divided and then perhaps multiplied, depending on your strategy. Using what you know about fractions could have helped. A number of you sent the answer that Peter was short of some of the ingredients. There were two ways that the problem was solved and fully explained. These mathematicians came up with the following solutions and used their knowledge of decimals.
Ruth who lives in Gloustershire, decided that the answer to the Pumpkin Pie Problem is, he hasn't got enough ingredients to make two pies. She got the answer by dividing the ingredients by 80 and multipliyingby 2. Both Helena from Tattingstone School, and George of Rosebank Primary School, Leeds, divided mother's recipe by 40. Why do you think they decided to do this?
Here's Ruth's solution, she did an excellent job of setting out her calculations in a chart.
| 80 pies take | What he needs | What he has got |
|---|---|---|
| 120 Eggs | 3 eggs | 2 eggs |
| 27L Milk | 0.675L Milk | 0.66L Milk |
| 480 tbs. Sugar | 12 tbs. Sugar | 15 tbs. Sugar |
| 100 teasp. Cinnamon | 2.5 teasp Cinnamon | 1.5 teasp Cinnamon |
| 140 cups of Pumpkin | 3.5 cups of Pumpkin | 4 cups of Pumpkin |
Helena and George both provided clear explanations. George explained: "To find out if Peter had enough to make 2 pies I divided each of the ingredients by 40. For the 10 dozen eggs I first multiplied 10 by 12, because dozen means 12. Then I divided the answer by 40.
10 X12 = 120.
120 / 40 = 12 / 4 = 3
Peter had 2 eggs but he needed 3 eggs so he had to buy 1 more egg.
To find out how many litres of condensed milk was needed, I first converted the litres of condensed milk into millilitres of condensed milk. Then I divided the answer by 40 again.
27litres X 1000 = 27,000ml
27,000ml / 40 = 675ml
Peter had 666ml but he needed 675ml so he needed to buy 9ml of condensed milk.
To find out how many tablespoons of sugar Peter needed I just divided the amount for 80 pies by 40.
480 / 40 = 12 tablespoons of sugar.
Peter had 15 tablespoons of sugar so he had 3 more.
To find out how many teaspoons of cinnamon, I divided the original amount again by 40.
100 / 40 = 10 / 4 = 2.5 teaspoons of cinnamon.
Peter had 1.5 teaspoons of cinnamon, so he needed to buy 1 more teaspoon of cinnamon.
Finally, to find out how many cups of pumpkin were needed I divided 140 by 40.
140 / 40 = 14 / 4 = 3.5 cups of pumpkin.
Peter had 4 cups of pumpkin so he had 0.5 cups of pumpkin more than he needs".
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
Ratio is a notoriously difficult topic and many teachers avoid it! There are many possible ways of solving this problem which make it ideal for class or group discussion, and offer opportunities to assess your children's understanding and misconceptions. You will need to allow time for a reasonable discussion.
Possible approach
Working in pairs gives the children opportunities to clarify their thinking. Give each pair a large piece of paper on which to record - many may want to draw pictures and working on small pieces of paper isn't helpful. Having explained the problem, provide each pair with the two lists.
Give a little time for the children to 'get into' the problem. When appropriate, draw the group or class together and ask what they have found useful to begin. Give more time for them to use some of these ideas, or continue with their own.
As each pair comes to a solution, ask them to prepare a new piece of paper which will help them to explain their working. Choose a confident pair to start and then invite each pair in turn to explain what they have done. The rest of the group can ask questions if clarification is needed.
Focus the children's attention on the different ways pairs have solved the problem but how all involve scaling amounts up and down - ratio and proportion.
Key questions
Can you paraphrase the question (ask it in a different way) to make it easier to understand?
What do we need to find out?
Possible extension
Most children will have focused on the eggs first as these are easier to calculate than litres, teaspoons etc. The question is then answered as no matter how much of everything else he has, he doesn't have enough eggs. Children who find this quite easy could be challenged to find out what additional other ingredients would have to be bought for two pies. What is the largest number of pies he could make without buying more of each ingredient?
Possible support
This is probably not a suitable question for children who struggle with the idea of doubling, halving and other simple ratio.