At a bake sale, Scarlett and Sophia are selling slices of apple pie and lemon pie packaged together. They have 90 slices of apple pie and 80 slices of lemon pie, and they want to sell all the pie slices. What is the greatest number of packages they can put together? How many slices of each type of pie are in a single package?
This question is quite wordy. So, we’re going to go through it again and highlight the important information and check that we understand it. The problem takes place at a bake sale where Scarlett and Sophia are trying to sell slices of apple pie and lemon pie. And we’re told that they package them together.
The first important piece of information that we’re told is that they have 90 slices of apple pie. We’re also told that they have 80 slices of lemon pie. And the next piece of information is easy to miss, but it’s very important. The girls want to sell all of the pie slices. In other words, the 90 slices of apple pie need to be split up into equal groups, there needs to be nothing left over. And the same for the 80 slices of lemon pie. They also need to be split up into equal groups with nothing left over.
Our question asks us, what is the greatest number of packages they can put together? Let’s start by thinking about our 90 slices of apple pie. What packages could we put together? Well, we could make one large package that has 90 slices in it. Or we could have 90 tiny packages with one slice in them. We know this because one times 90 is 90. Also, 90 times one is 90. Can you see what we are doing here? We’re finding the factors of 90, the numbers that we can split 90 into without there being anything left over. So, what are the other factors of 90? What multiplications do we know that equal 90?
Two lots of 45 make 90. Three lots of 30 equal 90. 90 is not in the four times table, but it is a multiple of five. We know 20 fives are 100. So, 90 is two less fives than that. Five multiplied by 18 equals 90. Because three times 30 equals 90, we also know six lots of half of 30 make 90, so six times 15. And finally, the one we learn when we learn our times tables facts, nine lots of 10 make 90. So, the factors of 90 are one, two, three, five, six, nine, 10, 15, 18, 30, 45, and 90. These are all the amounts that we could split the 90 slices of apple pie into. For example, we could make nine packages with 10 slices in, or maybe five packages with 18 slices in.
But the question tells us that the girls are selling their slices of apple pie and lemon pie together. So, we need to have the same number of packages. There are only 80 slices of lemon pie. How could we split up the number 80? We’re looking again for the factors of this number. One times 80 equals 80. So again, we could split the lemon slices into one large package that contains 80 slices or 80 tiny packs with one slice in. What else makes 80?
Two lots of 40. Four lots of 20. We know that eight 10s make 80. And so, the number of fives must be double eight. 16 fives make 80. So, the factors of 80 are one, two, four, five, eight, 10, 16, 20, 40, and 80. Because Scarlett and Sophia are selling the slices of apple pie and lemon pie together, they need to split them into the same number of packages. So, which numbers do they have in common? What common factors are there between 90 and 80? One, two, five, and 10. The greatest number of packages they can put together is 10. 10 is the greatest common factor between 90 and 80.
Now, we can answer the second part of our problem. How many slices of each type of pie are in a single package? Well, if we know there are 10 packages, all we have to do is to use division to find the answer. 90 slices of apple pie divided into 10 groups, or 10 packages, equals nine slices in a group. And 80 divided by 10 equals eight. So, 80 slices of lemon pie divided into the 10 packages means they’ll be eight slices of lemon pie in a pack. The greatest number of packages that girls can put together is 10. And in each package, there’ll be nine slices of apple pie and eight slices of lemon pie.