Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem5

Video Transcription

Question number 5. Each year on her birthday, Anna receives a number of roses equal to her age. She always keeps all her roses and currently has 120. How old is Anna?  We recall the formula for adding up the consecutive integers we used already in question number 3.  And that formula goes like this. If I add 1, plus 2, plus 3, plus 4, all the way up to some number n,  then the answer can be quickly computed as that number n times n plus 1, and then we divide by 2.  And each term here in the sum would represent the number of roses Anna received on one of her birthdays consecutively  from 1 all the way up to her current age. And so we know that 120, that's the number of roses, and that is equal here to n times n plus 1 divided by 2. And then we can solve for n.  So multiplying both sides by 2, we have 240 is equal to n times n plus 1. And so what we have here on the right hand side is a product of consecutive integers.  And we have to find an integer and the next integer such that they multiply to 240. So how do we factor 240? Well, that would be, for example, 24 times 10, and 24 is 8 times 3,  so 2 to the 3rd times 3, 10 is 2 times 5, and so we have 2 to the 4th power times 3 times 5.  And now we need to factor into consecutive integers. So, for example, just grouping these together,  2 to the 4th power is 16, and 3 times 5 is 15. That tells us that n is equal to 15, so n plus 1 is equal to 16.  And we have found that number we were looking for, such that 1 plus 2 plus 3 all the way up to 15  gives us the desired sum of 120. So that is Anna's age. She is currently 15 years old.

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