Hello everyone, this is problem four of the grade 11 and 12 2025 math kangaroo test. So, in what interval does the value of 88 times 888 lie? Our first answer choice is between 8 and 88. Our second, B, is between 88 and 888. C, between 888 and 8888. D, between 8888 and 88888. And finally, E, between 88888 and 888888. I apologize for all that. And now I recommend that you try the problem by yourself. Okay, so there are a few ways to go about doing this. We could just actually just straight compute that answer. But that seems kind of like a lot of work. And, you know, mathematicians are lazy. I don't want to have to do that work. So what I'm going to do instead is I'm going to try and make an upper and a lower bound. And then sort of see where my answer lies. So let's start with our upper bound. So upper. So we know that the value of 888 times 888 is going to be less than or equal to 100 times 888. And multiplying by like exponents of tens is pretty easy. This is just 8 8 8 0 0. Which means that like off rip, E is not a valid answer choice. Because 888 0 0, which is necessarily greater than or equal to our solution, is like, it's not included in this bound. So we can we can immediately cut that out. And now we need a lower bound. And this lower bound can be obtained by just sort of cutting off these final eights in the equation. So we can lower bound it with 80 times 800. And again, multiplying by exponents of like 10 is kind of easy. So this becomes 64 with three zeros in the end. So 64,000. And then we can now remove A, B, and C as bounds. We don't like them at all. Because 64,000 is, you know, greater than each of these upper bounds. So, you know, they're not valid. And that leaves us with only the answer choice of D. Between 8888 and 88888.

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