News from Mathnasium of Fort Lee
The Hot Dog Conundrum
Jul 14, 2016
Happy Word Problem Wednesday! This week, a backyard barbecue staple is in the spotlight. Solve our hot dog word problem and check the answer below!
Hot dogs come in packages of 10. Hot dog buns come in packages of 8. Each bottle of ketchup has enough ketchup in it for 24 hot dogs. What is the least number of packages of hot dogs, packages of hot dog buns, and bottles of ketchup you can buy so that you have the same number of servings of each?
Here's the solution!
There are two mathematical topics taught around the 4th grade that are often forgotten: the Greatest Common Factor (GCF) and the Least (or Lowest) Common Multiple (LCM). These are usually taught as methods for solving fractions but it's easier to apply other strategies and so these fundamental concepts are not exercised enough.
Many of our students have to be re-taught these topics. Often all they remember are the acronyms and nothing of meaning. I always start by asking the student to say the words “Greatest Common Factor” and “Lowest Common Multiple” and explain to me the meaning of each word. After that, it is easier for them to “get it”.
- Greatest – means larger of two or more things.
- Common – means the same.
- Factor – whole numbers that divide evenly into a number, for example, factors of 8 are 1, 2, 4, and 8, each of which will divide 8 into 8 whole pieces, 4 whole pieces, 2 whole pieces, and 1 whole piece respectively.
- Least – or lowest, means the smaller of two or more things.
- Common – means the same
- Multiple – one or more of something, for example, multiples of 8 are 8, 16, 24; that is, one 8 (8x1), two 8s (8x2), three 8s (8x3), and so on.
So, the GCF of two or more numbers is the largest valued factor that are common to both numbers. For 8 and 12, this would be 4.
And the LCM of two or more numbers is the smallest valued multiple that are common to both numbers. For 8 and 12, this would be 24.
Now that we’ve got those two very closely related ideas differentiated, then answer to this puzzle is to apply the least common multiple (LCM) concept since we need to buy the least number-of-whole hot dog packages, bun packages, and bottles of ketchup.
The least common multiple of 10, 8, and 24 is 120 which is the smallest number of complete hot dogs consisting of 1-bun, 1-hot dog, and 1-squirt of ketchup. The least number of packages of each you can buy so that you have the same number of servings of complete hot dogs are: 12 packages of 10-hot dogs makes 120 hot dogs, 15 packages of 8-buns makes 120 buns, and 5 bottles of 24 servings makes 120 squirts of ketchup.
There are standard algorithms that make finding the LCM and CFG a snap. Befittingly for this "foody" article, one of them is called the cake method. Come in and ask us about it. For many readers, this is likely the first – and often last – practical application of the LCM! We think the insight is surprisingly fun, don't you?
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