Themed Challenge 4 (Number Theory): Miracle Age Potion!
The following is a company's advertisement for an age potion:
"Do you feel like your wonder years are behind you? Do you get constantly reminded that you aren't as young as you used to be? Do you feel yourself helplessly attempting to make yourself feel younger?
Well, we have a solution! Try Company A's new miracle age potion! It really works listen to these satisfied customers.
Customer A: I used to be years old, but thanks to Miracle Age Potion, I have become times younger!
Customer B: I used to be years old, and my friend was , but after taking Miracle Age Potion, I felt like a year old, and he became times younger! This product changed my life
Customer C: I used to be years old, and I used to feel helpless, then after taking Miracle Age Potion, I was as spritely as an year old. I became over times younger.
That's not all! Miracle Age Potion is safe on kids, aged , as their age doesn't even change!
So try Miracle Age Potion today! They say age is but a number. We are here to reduce that number
Catchy Jingle "
Now, say people, aged differently, under years of age, take the potion. It is found that all of them become times younger. Find their average age.
Details and Assumptions:
The Miracle Age potion takes a certain person's age and puts it into a function to obtain their new age. You will need to find this function from the clues given.
If someone becomes
times younger, it means that their age is divided by
.
The answer is 30.
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The miracle age potion changes your age to the sum of your original age's digits: 4 5 becomes 9 , 4 9 becomes 1 3 , 5 0 becomes 5 , 9 9 becomes 1 8 and all integers from 0 to 9 remain unchanged.
Let the original age be 1 0 a + b , where a , b are respectively the tens and ones digit of the four people's ages. We wish to find the four pairs ( a , b ) that give us 1 0 a + b = 4 ( a + b ) or equivalently 2 a = b . We then get the two-digit numbers 1 2 , 2 4 , 3 6 , 4 8 , all of which happen to be divisible by 4 . The average of these numbers are then 3 0