Question 190610
# 2

Q: <font color=red> Make a conjecture about the relationship between the original number and the final number in the following process.

Pick a number.
Multiply the number by 5
Add 10 to the product
Divide the sum by 5
Subtract 1 from the quotient  </font>


A:

Let "x" be the unknown number


Example # 1


Step 1) "Pick any number". I choose the number 3


Step 2) "...multiply the number by 5". 3*5 = 15


Step 3) "Add 10 to the product". 15 + 10 = 25


Step 4) "Divide the sum by 5". 25/5 = 5


Step 5) "Subtract 1 from the quotient". 5 - 1 = 4



So after picking the number 3, the result after all of these operations is 4



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Example # 2


Step 1) "Pick any number". I choose the number 13


Step 2) "...multiply the number by 5". 13*5 = 65


Step 3) "Add 10 to the product". 65 + 10 = 75


Step 4) "Divide the sum by 5". 75/5 = 15


Step 5) "Subtract 1 from the quotient". 15 - 1 = 14



So after picking the number 13, the result after all of these operations is 14



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Example # 3


Step 1) "Pick any number". I choose the number 21


Step 2) "...multiply the number by 5". 21*5 = 105


Step 3) "Add 10 to the product". 105 + 10 = 115


Step 4) "Divide the sum by 5". 115/5 = 23


Step 5) "Subtract 1 from the quotient". 23 - 1 = 22



So after picking the number 23, the result after all of these operations is 22


Are you starting to see the pattern?



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Example # 4


Step 1) "Pick any number". I choose the number 9


Step 2) "...multiply the number by 5". 9*5 = 45


Step 3) "Add 10 to the product". 45 + 10 = 55


Step 4) "Divide the sum by 5". 55/5 = 11


Step 5) "Subtract 1 from the quotient". 11 - 1 = 10



So after picking the number 9, the result after all of these operations is 10



Notice how whatever number we choose, the result is simply the next number. 



Conjecture: If our number is "x", then the result of these operations is {{{x+1}}}



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Here's what's happening algebraically:


Step 1) "Pick any number". I choose the number "x"


Step 2) "...multiply the number by 5". x*5 = 5x


Step 3) "Add 10 to the product". 5x+10 


Step 4) "Divide the sum by 5". {{{(5x+10)/5=(5(x+2))/5=(cross(5)(x+2))/cross(5)=x+2}}}


Step 5) "Subtract 1 from the quotient". {{{(x+2)-1=x+2-1=x+1}}}



So after choosing the number "x", the result is {{{x+1}}}