How would I explain this problem to middle school math students?

[HornetMathRIII]

1. pick a number between 1 and 10
2. multiply the number by 2
3. add 5
4. multiply by 50
5. add 1762 if you have already had your birthday this year
6. add 1761 if you have not already had your birthday
7. subtract the year of your birth

The resulting number is the first number you picked and your age. i.e. 213 or 714

How does this work?

x is an element of {1,2,3,4,5,6,7,8,9,10}
where a = 1762 if you have already had your birthday this year, where b = 1761 if you have not already had your birthday this year
where cdef is your year of birth e.g. 1986

(2x + 5)
50(2x + 5)
50(2x + 5) + (a or b), where a = 1762 if you have already had your birthday this year, where b = 1761 if you have not already had your birthday this year
50(2x + 5) + (a or b) - cdef, where cdef is your year of birth e.g. 1986

I'm not really getting anywhere with my explanation or proof beyond this.

Any help is appreciated.

 

[DrPhil]

1. pick a number between 1 and 10
2. multiply the number by 2
3. add 5
4. multiply by 50
5. add 1762 if you have already had your birthday this year
6. add 1761 if you have not already had your birthday
7. subtract the year of your birth

The resulting number is the first number you picked and your age. i.e. 213 or 714

How does this work?

Expand the expression you have after step 4:
(2x + 5)*50 = 100x + 250
Note that 100 x is just the digit x written in the 100s position of the result

When you add 250 + 1762 you get 2012, which just happens to be the current year.
When you subtract your birth year from this year, you have your age at this year's birthday. By adding one less if you haven't had a birthday yet this year, you have your age today.
[Note that on or after January 1, you will have to add 1763 or 1762.]

 

[JeffM]

1. pick a number between 1 and 10
2. multiply the number by 2
3. add 5
4. multiply by 50
5. add 1762 if you have already had your birthday this year
6. add 1761 if you have not already had your birthday
7. subtract the year of your birth

The resulting number is the first number you picked and your age. i.e. 213 or 714

How does this work?

x is an element of {1,2,3,4,5,6,7,8,9,10}
where a = 1762 if you have already had your birthday this year, where b = 1761 if you have not already had your birthday this year
where cdef is your year of birth e.g. 1986

(2x + 5)
50(2x + 5)
50(2x + 5) + (a or b), where a = 1762 if you have already had your birthday this year, where b = 1761 if you have not already had your birthday this year
50(2x + 5) + (a or b) - cdef, where cdef is your year of birth e.g. 1986

I'm not really getting anywhere with my explanation or proof beyond this.

Any help is appreciated.

The problem is just a function of setting up definitions and doing a little algebra.

\(\displaystyle Let\ i\ be\ the\ integer\ chosen.\)

\(\displaystyle Let\ b\ be\ your\ birth\ year.\)

\(\displaystyle Let\ x = 0\ if\ your\ next\ birthday\ occurs\ later\ in\ 2012\ than\ today\ and\ let\ x = 1\ otherwise.\)

\(\displaystyle Your\ age\ in\ whole\ years\ (floor\ function) = y = 2011 + x - b.\) Make sure you understand why this is correct.

\(\displaystyle Let\ n = 100i + y.\)

\(\displaystyle \{(2i + 5) * 50\} + 1761 + x - b = 100i + 250 + 1761 + x - b = 100i + 2011 + x - b = 100i + y = n.\)