Okay, so I was messing around with some math problems the other day, and I came across this thing that got me kind of stumped. I had to figure out how to write the repeating decimal .318 (with the 18 repeating, you know, like .318181818…) as a fraction. Sounds simple enough, right? But it had me scratching my head for a bit.
So, first, I tried the usual trick of setting it up as an equation. I let x = 0.3181818… Then I multiplied that by 100 since the repeating part has two digits. That gave me 100x = 31.81818… Now here’s where it got a bit tricky.
I subtracted the first equation from the second one, like this: 100x – x = 31.81818… – 0.31818… This simplifies to 99x = 31.5. I was feeling pretty good about it until I hit this point. That 31.5 wasn’t exactly neat, you know? Not the clean integer I was hoping for.
I paced around for a bit, mulling it over. Then it hit me! I needed to get rid of that .5. So, I multiplied both sides of the equation 99x = 31.5 by 10. That gave me 990x = 315. Much better! Now, I could easily solve for x.
I divided both sides by 990, and boom! I got x = 315/990. I was excited, but of course, I had to simplify it. I started playing around with factors, trying to find common ones for both 315 and 990.
I noticed that both were divisible by 5. Dividing by 5 gave me 63/198. Tried dividing by 3 again, and got 21/66. And then I noticed they were still both divisible by 3, so one more time, divided by 3 and I finally got 7/22. That was it! The simplest form.
- Let x = 0.3181818…
- Multiply by 100: 100x = 31.81818…
- Subtract: 100x – x = 31.81818… – 0.31818… which simplifies to 99x = 31.5
- Multiply by 10 to get rid of the decimal: 990x = 315
- Divide: x = 315/990
- Simplify the fraction: 315/990 = 63/198 = 21/66 = 7/22
So, yeah, after all that, I finally figured it out. The repeating decimal .318 is equal to the fraction 7/22. It took some trial and error, but I got there in the end. It’s kind of fun to look back and see how I stumbled through it, right?
What I learned today? Sometimes you just gotta keep plugging away, trying different things until something clicks. And don’t forget to simplify!
