Grades 11-12 Video Solutions 2025-2025_11-12

Video Transcription

This is problem seven for 11th and 12th grade, 2025 on the Mac and Guru.  Mike obtains a number X by dividing the number square root of 11 by three. Where's the number X to be found on the number line?  We have A between zero and one, B between one and two, C between two and three, D between three and four, and E between four and five. Take a moment and try this one for yourself.  Now pause the video.  And awesome, okay.  So if you know really anything about radicals, or even if you don't know anything about radicals, that looks sort of like angry. We don't, like if I were to try to give you  a decimal expansion for that,  I would be sitting here forever because it is this thing called an irrational number.  And yeah, they go on forever. They don't have a like fractional thing we can work with. So what we're going to have to try to do is estimate. And for me, the best way to estimate  are to get some perfect squares because the square roots of perfect squares just become integers. So we have square root 11 here,  and we want to find a square root that's smaller than 11. Let's say nine or something. So square root of nine is less than square root of 11. And the next perfect square is 16.  So let's take square root of 16.  So we can be confident in this inequality, and then we can sort of bound it. We can divide everything by three to actually get X. So we get that X, because it's 11 over three,  is going to be less than root 16 over three, and it's going to be greater than root nine over three.  And, you know, rewriting everything, this becomes three over three less than X, less than four over three.  And now we can like just check all these things. This is one. This is, you know, 1.3 repeating, or if you want to keep it as four over three,  that's totally fine. But critically, these are like completely within the interval that B defines, between one and two. So that means if X is within that sort of nested interval,  then X should also be between one and two, unless we obtain the answer choice of B.

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