Grades 7-8 Video Solutions 2025-2025_7-8

Video Transcription

Problem number 25. The letters p, q, r, s, and t represent five consecutive positive integers, though not necessarily in that order. The sum of p and q is 69,  and the sum of s and t is 72. What is the value of r?  So if we have numbers that are close together, then we would expect these numbers to approximately  equal their average, plus or minus a few values. In this case, since we know that p and q  sum up to 69, the two numbers will be close to the average of 34.5, which is 69 divided by 2.  And in this case, that means that p and q must be 34, 35, or 33, 36. They can't be more than  four apart, since we know the whole range from p to t spans five numbers, which means that the  first number cannot be more than four away from the second. By the same logic, we can start to  figure out what s and t can be. They must be close to their average of 36, but they can't  go to equal 36, because again, they have to be distinct. So this means that s and t are 35 and  37, or 34 and 38, and again, we don't know the order. But now we can start to figure out what  values could work. If p and q are 34 and 35, then neither of the options for s and t works.  For example, whatever is 34 will clash with the option for 34 or 38, so s and t cannot be 34,  38. But they also clash with the other option for 35, 37, because we already said one of p and q is  35. So neither option for s or t works, and so that means that p and q cannot be 34 and 35.  This means that p and q must be 33 and 36. Now the largest possible number we can now have is 37.  Since 33 is in the list, we can't have a number that's more than four away from 33. In particular,  we cannot have 38, since 33 is five away from 38. That means that s and t must be 35 and 37.  And so since we have all the values p, q, s, and t in 33, 36, 35, 37, the missing number, which is r, must be 34. That means the answer is c, 34.

Video Summary

Summary Not Available