Problem Solving Using Venn Diagrams

This trick or trap comes up a lot in combinatorics problems.It’s called “double counting.” To avoid this problem we can use Venn diagrams.This module contains three lessons that are build to basic math vocabulary.

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MULTIPLES OF 3 AND NOT 5 OR 7: The number of elements in A = the integer part of (1000/7) – (9 19 38) = 142 – 66 = 76 Our final diagram: So the number of elements less than 1000 which are divisible by 3, 5 or 7 = 248 96 76 19 57 38 9 = 543 And the number of elements less than 1000 which are NOT divisible by 3, 5, nor 7 = 1000 – 543 = 457 The answer is (A).

Keep in mind I’m going to leave you with some challenging problems to try on your own.

All of these problems can be solved with Venn diagrams 1.

How many positive integers less than 1,000,000 are neither squares nor cubes? Given a random 6-digit integer, what is the probability that the product of the first and last digit is even?

Let’s agree the circle labeled A represents all the numbers less than 1000 which are divisible by 3.

The circle labeled B represents all the numbers less than 1000 which are divisible by 5.Let’s move from top to bottom: MULTIPLES OF 3 AND 5: Multiples of 3 and 5 are multiples of 15 The number of elements in A = the integer part of (1000/21) – 9 = 47 – 9 = 38 Now our diagram looks like this: Almost there…remember not to double count the portions of the diagram that are already labeled!Three worksheets to practice working with Venn Diagrams included in higher GCSE (9-1) examination.(Can be used with all boards, but questions taken from Edexcel or IB papers) 1.Because 3, 5, and 7 don’t share any factors, their least common multiple is the product of 3, 5, and 7: 105.To find the number of multiples of 105 less than 1000, we divide 1000 by 105.Problem Mat - print out on A3 double sided and you have a collection of Venn diagram questions students can attempt.Front page are simpler questions, with more challenging questions overleaf. Three past paper questions for students to attempt with solutions. A graduated mixture of past paper questions (levelled from warm to hot) plus a timed challenge of 5 questions to attempt.We have to careful about integers that are divisible by more than one of our numbers.For example, 21 is divisible by 3 and 7, but we can only count it once.

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