In the most recent Math Fundamentals Problem of the Week, *Peanut Butter and Jelly*, students were told that a 48 ounce jar of jelly makes 60 sandwiches and that a 16 ounce jar of peanut butter makes 12 sandwiches (among other things). The Extra question asked whether each sandwich would contain more peanut butter or more jelly. Much like the *So Many Salmon* problem, this question can be answered in a number of different ways.

One method involved figuring out how many ounces of peanut butter or jelly are required to make a particular number of sandwiches. (The answer to the main part of the problem was 120 sandwiches, so it makes sense that some students would figure out how much peanut butter and jelly was used to make those 120 sandwiches.)

Breanna, East Middle School:If I combine 3 regular 16 ounce peanut butter jars together, it will be equal to 1 regular 48 ounce jar of jelly. Now I will combine each 3 peanut butter jars together so that both peanut butter and jelly jars will have equals amounts of 48 ounces in them. So, if I try to find if a sandwich has more peanut butter or jelly, the jars will have the same amount in them.Now to try to figure this question out, I have to take the info. I had before and change it so that it fits with our new amount in the peanut butter jars.

Two 48 ounce jars of jelly= 120 sandwiches

≈ three 48 ounce jars of peanut butter= 120 sandwiches</pSo now to make a sandwich I know that it takes more peanut butter than jelly because it takes approximately 1 more 48 ounce jar of peanut butter to make 120 sandwiches than it does jelly.

Samantha, Birch Wathen Lenox School:16 oz of peanut butter is needed to make 12 sandwiches (16/12) and converted it to 80 oz for 60 sandwiches (80/60) in order to compare to 48 oz of jelly making 60 sandwiches (48/60). So there is more peanut butter per sandwich.

Eric, Sacajawea Middle School:There is more peanut butter than jelly in each sandwich because 1 jar of jelly makes 60 sandwiches and 5 jars of peanut butter makes 60 sandwiches. The peanut butter jar is 16oz and the jelly jar is 48oz. so if you multiply 16oz by 5, it will become an equal comparison and the larger number has more of it per sandwich. So 16oz x 5 =80oz 80oz is greater than 48oz so there is more peanut butter per sandwich.

Other students figured out how many sandwiches can be made with a particular amount of peanut butter or jelly.

Twentyseven, Caughlin Elementary School:A 48 ounce thing of peanut butter will make 36 sandwiches and the same amount of jelly makes 60

The most popular method was to figure out exactly how much peanut butter and how much jelly are in a single sandwich.

Cat, Mount Desert Elementary School:For the peanut butter, I divided 16 ounces by 12 sandwiches because there are 16 ounces in a jar of peanut butter that makes 12 sandwiches. The product was 1.3 ounces of peanut butter repeating. For the jelly, I divided 48 ounces by 60 sandwiches because there are 48 ounces in a jar of jelly that can make 60 sandwiches. The product was 0.8 ounces of jelly. 1.3>0.8 therefore, there is more peanut butter in each sandwich.

Emma, Sacajawea Middle School:Each sandwich would contain more peanut butter. Since each jar of peanut butter contains 16 ounces of peanut butter, and makes 12 sandwiches, you do 16/12, which is 1.33 ounces. You do the same with the jelly. 48/60=0.8 ounces. (48 ounces in 1 jar of jelly/60 sandwiches.) So that means that each sandwich would contain more peanut butter than jelly.

Michael, Rosemont School of the Holy Child:Each sandwich will contain more pb than jelly because each sandwich will have 4/3 oz of pb (16 oz pb/12 sand=4/3 oz per sand) and 4/5 oz of jelly (48 oz jelly/60 sand=4/5 oz per sand).

Angi, Hanover Street School:Each sandwich will have more peanut butter because there is 75oz. in each sandwich of Peanut Butter. In a jar of jelly there is 8oz. in a sandwich. I did 48oz ÷60=.8oz (jelly)then I did 16oz ÷12oz=.75oz

Each of the methods above required some calculations. Nothing wrong with that, but do we actually need to do any calculations to answer this question? Consider these two explanations:

G, McDougle Elementary School:A 48 oz. of jelly makes 60 sandwiches. Because 48 is less than 60, you must use less than an ounce of jelly on each sandwich. A 16 oz. jar of peanut butter makes 12 sandwiches. Since 16 is greater than 12, you use more than ounce on each sandwich. So there must be more peanut butter than jelly on each sandwich.

Shawn, Lakewood Elementary School:There will be more peanut butter in 1 sandwich because 1 peanut butter jar contains 16 ounces but can only make 12 sandwiches, which means that there is more than 1 ounce of peanut butter in a sandwich because 16>12. But a jelly jar holds 48 ounces but can make 60 sandwiches, so that means that there is less than 1 ounce of jelly in a sandwich because 48<60.

Many comparisons don’t require calculations. Understanding such situations makes students more agile as mathematicians. They learn to think about what they really need to know in order to answer a question, and tend to ponder situations a little more before diving in and making calculations. How did your students solve this question? If they solved it making calculations, what do they think of the last two solutions? Let us know!

Some ** Peanut Butter and Jelly** links in case you are interested:

- The problem [requires a Math Forum PoW Membership].
- Information about accessing “Peanut Butter and Jelly” (and a selection of all our PoWs) for 21 days with a free Math Forum trial account.
- Information about becoming a Math Forum Problems of the Week Member. Consider starting with a $25 membership, which gives you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!