Challenge 6 - Deli(cious) Sandwiches
Challenge 6 is all about finding different combinations.
Maths teacher Chris Smith and pupils from Grange Academy are here to explain.
The Maths Week Scotland Daily Challenges have been set by the Scottish Mathematical Council.
Teacher: This problem is all about finding different combinations.
A corner deli stocks 4 kinds of bread, 5 kinds of meat, and 3 kinds of cheese. It sells three types of sandwiches.
One has one type of bread and one type of meat.
The second has one type of bread and one type of cheese, and the third has one type of bread with one type of meat and one type of cheese.
How many different sandwiches are on sale?
Pupil: Think about the three different types of sandwiches one at a time.
If you look at how many different types of bread you have and how many bits of meat you have, then you can see the different combinations.
Pupil: You can draw a table for every type of filling for each type of bread.
Pupil: You could try working backwards by working out the different fillings first.
Teacher: 4 times 5 equals…
Pupil: Good luck.
So here's the challenge:
A corner deli stocks four kinds of bread, five kinds of meat and three kinds of cheese.
It sells three types of sandwiches:
One has one type of bread and one type of meat
One has one type of bread and one type of cheese
And one has one type of bread with one type of meat AND one type of cheese.
How many different sandwiches are on sale?
Need a hint?
- Think about the three different types of sandwich one at a time
- Could drawing a table for each type of bread and filling help?
- You could try working backwards by working out the different fillings first
Solution
Worked out the answer? Here's how you can do it.
Did you work out how many different sandwich combinations are on sale? Here's how we got our answer.
Let's work through each of the three types of sandwich combinations in turn.
There are four different types of bread and each can have five different types of meat. Four times five equals 20 sandwiches.
Each of the four types of bread could have one of the three different cheeses, four times three equals 12 sandwiches.
The last combination could have any of the four types of bread filled with one of the five meats, and one of the three cheeses, four times five times three equals 60 sandwiches.
So if we add all of these up 20, add 12 and 60 gives us a total of 92 different sandwich combinations.
Well done if you got that one, without getting hungry.
Step 1
Let's work through each of the three types of sandwich combination in turn:
- Bread and meat.
There are four different types of bread and each can have five different types of meat.
4 x 5 = 20 sandwiches
Step 2
- Bread and cheese.
Each of the four types of bread could have one of three different cheeses.
4 x 3 = 12 sandwiches
Step 3
- Bread and meat and cheese.
The last combination could have any of four types of bread, filled with one of five meats, AND one of three cheeses.
4 x 5 x 3 = 60 sandwiches
Step 4
Now we can add the number of different sandwiches for each type together.
20 + 12 + 60 = 92
The deli serves a total of 92 different sandwich combinations.
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