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JH
Numerade Educator

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Problem 52 Medium Difficulty

Express the number as a ratio of integers.
$0 . \overline{46}=0.46464646 \ldots$

Answer

$0.464646 \cdots=\frac{46}{99}$

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Video Transcript

Let's go ahead and express this number as a ratio of injures. So what we'd like is something of the form. Like peeled or cube. We're P humor injures. So these are numbers like like, for example, negative. Three negatives. Who? Zero and positive numbers and someone. So no fractions, no pies or anything like that. So what we could do here is we can rewrite this as forty six over ten square, So first, you know, let me actually take us that back here. First, let me rewrite this decimals or something. Someone to play for six. Then, since I already took out the first four six year, I'LL put two zeros in front for the next four six. So that accounts for the next term and so on. So the next time I put four zeros because the first four spots are already taken for six year for six here in someone. So each time What what am I doing with the decimal East? Time decibel moves two to the left and equivalently we could think of. This is each time we go from one term to the next time we are more supply lines. Bye. One over tents were so we can write. This is forty six over one hundred and then forty six over. Then we multiplied by another ten squared on the bottom again that this is equivalent of pushing the decimal from here to two places to the right, to the left. Excuse me. This is why we include another to here that gives us the four and same thing over here for the next one. I want forty six. But this time I want another two zeros from the previous one. So I have two more zeroes here, so I have to multiply by another one divided by ten square. So that gives me two more tens on the bottom in someone. And if you like, you could even pull out the four. The forty six you have won over ten square one over ten to the fourth one over ten to the sixth and so on. And what are we multiplying by Each time soldiers are for this geometric series? Whoa! From gets from one serves in the next were also were always multiplying by one over Since we're so we know that this our satisfies this inequality here. So we know that it converges. I mean, this is a number. This is a real number. So the Siri's must converge. Now let's use the formula for the geometric series to find the sum. So recall the formula. It's first term and let me go back and make a common here. This was actually a necessary Let's just stick with this right now. Now we know the fur Geometric series. You do the first term over one minus the R in our problem, the way that this is written in the first term, forty six all over sense where that's our very first term. And then we do one minus are there. But our is just won over a hundred. So we have forty six over a hundred, and then we have ninety nine over hundred. The denominator. Let's cancel those one hundreds, and then we just get forty six over ninety nine. And that's precisely what we wanted. This's our ratio of imagers and the two in injures here, or forty six and ninety nine, and that's our final answer