Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem20

Levels 9&10 Video Solutions 2014 problem20

Video Transcription

Question number 20. Let p, q, and r be positive integers and such that p plus this fraction 1 divided by q plus 1 over r is equal to 25 over 19. The question is  which of the following is equal to the product of p, q, and r? So let's look at our equation here. We have that 25 divided by 19, which is in lowest terms,  is equal to this expression, which we should simplify until we have a numerator and a denominator, no compound fractions, because 25 over 19 being in  simplest terms will just allow us to say that the denominator is 19 and the numerator is 25. So first let's work with the second term and multiply q by  r plus 1 over here. The common denominator would be r and then we can invert and multiply and we see that this is p plus r over qr plus 1. Now it's  easy to get a common denominator. We see that 25 over 19 is equal to the product p times qr plus 1 plus r divided by qr plus 1. Okay, so now we see that since  25 over 19 is in already lowest terms, we see that qr plus 1 has to be equal to the denominator on both sides of the equation or the product qr is equal to  18. Okay, and now we have to just figure out what the value of p is and multiply by p on both sides here. So if qr is equal to 18 and p, q, and r are positive  integers, we see that r has to be in the set of positive integer divisors of 18. So r could be 1, r could be 2, r could be 3, r could be 6, r could be 9, or it  could be 18. And then we look at the numerator and we see that 25 has to be equal to p times qr plus 1 plus r, but qr here is equal to 18, so we will replace  this with p 18 plus 1 plus r, so 19p plus r has to be equal to 25. And this tells us that p has to be equal to 1 because if p is greater than 1, all of  these numbers being integers with p equal to 2, we already have to have a negative value for r, else r is negative, which we cannot have. So p is equal to 1  and we can just finish solving here. If p is equal to 1, then 25 is equal to 19 plus r, and r is equal to 6. So this is the value of r here that we had to  choose, but we already know that p is equal to 1, we know that q times r is 18, so multiplying that on both sides, we see that pqr is also 18. And so that is the  answer here, which comes out to be letter C.

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