Learning Objectives

# Problem 1 of 3

Prove, by contradiction, that √5 is irrational.

**Solution**

To solve this we will have to use the contradiction, which means: we have to start from the conclusion expected to the thesis.

Therefore:

am doing the images for this right now

# Problem 2 of 3

The Pensar family is looking for a nanny who is fluent in Portugese as well as English because Mrs. Pensar's work with the World Bank will take them to Portugal in two years. They would also like a nanny who can work extended evening hours and be available some Saturdays. The placement service director informs them that, of the 1000 nannies currently available,

• 399 speak Spanish

• 57/200 are available for extended evenings

• 38 are willing to work weekends

• 2.2% speak Spanish and work weekends

• 2/125 work extended evenings and weekends

• 6% speak Spanish and work extended evenings

• 4 meet all three requirements

After interviewing the four who met all three requirements, the Pensars decided they needed to meet a few more nannies. Thus, they agreed to drop the weekend requirement.

(a) How many more choices do they now have?

(b) How many applicants didn't meet any of their initial requirements?

**Solution**

From the last clue we can insert a four into the 3 overlap section because there are 4 people who have all three requirements.

Now we can insert 56 into the spanish-evening section because the number of nannies that have those requirements is 6% of 100 which is 60 but 4 is already in the center section so 60-4=56. As for the 22, 2.2% of 1000 is 22 and since there is 4 in the center the 22 becomes an 18.

As you can see I've added 16 and 209. I got the 16 from the 5th clue and it states that 2/125 nannies which mean 16 if you turn the fraction to 16/1000. The same goes with the 2nd clue, just turn the fraction into 285/1000 which means 285 nannies work evenings and then subtract the amounts of the overlaps that have to do with evenings from the total we just found and that is the number of nannies that work only evenings

Now we can complete the venn diagram by inserting the last numbers which are 0 and 376. To get zero you look at the 3rd clue and see that 38 nannies can work weekends and if you subtract the other numbers that have to do with weekends from 38 you get 0. So there are no nannies that work only weekends. The leftovers or the nannies that have none of these requirements is 3769
(1000-all numbers on venn-diagram=376)

a)How many more choices to they now have?
From the end of the story it says that the family dropped the weekend requirement and now they are left with 2 types of requirements and now they want the nannies that can both speak spanish and work evenings after the 4 nannies that have all 3 requirments and from the diagram it shows that a total of 56 people have those requirements

b)How many applicants didn't meet any of their initial requirements?
From the venn-diagram we can see that 376 nannies didnt meet the requirements because the 376 is on the outside of the venn-diagram

# Problem 3 of 3

coming soon ...

**Solution**

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