Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem 13. Kate and Tom are celebrating their birthdays today. Tom notices that one-nineteenth  of Kate's age is equal to one-seventeenth of his age. The sum of their ages is greater than 40 and less than 100. How old is Kate? So, to start off, let's set variables. So  let Kate's age be K and Tom's age be T. Now, the information in the problem says that one-nineteenth of K is one-seventeenth of T. And if we multiply both sides by 19 and  17, this tells us that 17K is 19T. Now, clearly the right side divides by 19, so this means  that the left side of this equation also has to divide by 19. But since 19 and 17 are both prime and don't share any common factors, this means that K is divisible by 19. And  similarly, T has to be divisible by 17. So, what does it mean to be divisible by a number? It means that we can find some X and Y so that K is 19X and then T is 17Y. So then,  if we substitute these new representations of K and T back into our original equation, we have that 17 times 19 times X is 19 times 17 times Y. And so this tells us that X is  equal to Y. So now, the sum of Kate's and Tom's ages are 17X plus 19X, which is 36X. And we know that this needs to be greater than 40 and less than 100. And we can quickly  see that the only way this works is if X is equal to 2. If X is 1, 36 is less than 40. And if X is 3, then 108 is more than 100. So this means that X has to be 2. And then  Katie's age is 19 times 2, which is 38. And so our answer is C.

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