Converting Repeating Decimals into FractionsDate: 06/25/2008 at 13:02:11 From: Sarah Subject: Converting repeating decimals into fractions How do you convert repeating decimals (in which only one number at the end is repeating) into fractions? ex. 0.91666666666666... ex. 0.38888888888888... ex. 0.84733333333333... I know how to do a single number repeating decimal like 0.6666... by multiplying by 10 and subtracting: 10N = 6.6666... N = 0.6666... --------------- 9N = 6.0000... N = 6/9 or 2/3 But I am not able to convert repeating decimals where there are other numbers before the part that repeats. There is only one repeating number, so I thought you would have to multiply it by 10, but it doesn't give the right answer. 0.9166666 x 10 = 9.16666, then I don't know what to do with a decimal (how to put it as a numerator) or 0.9166666 x 1000 = 916 916 --- , but that's the wrong answer. 999 Date: 06/25/2008 at 14:57:33 From: Doctor Achilles Subject: Re: Converting repeating decimals into fractions Hi Sarah, Thanks for writing to Dr. Math, and for explaining your thinking so clearly. You are right about how to do a single-digit repeating decimal by multiplying by 10. So to solve 0.7777777.... you know how to come up with 7/9. Let's look at a couple of similar problems. If you had 6.7777777..... you could think of it as 6 + 0.7777... so you would have 6.7777... = 6 + 0.7777... = 6 + 7/9 = 6 7/9 or 61/9 depending on if you wanted a mixed number or an improper fraction. Can you use similar thinking to try these two: 0.0777777... and 0.6777777... ? Can you please give that a shot and write back and show me your work? Good luck! - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ Date: 06/29/2008 at 19:16:26 From: Sarah Subject: Converting repeating decimals into fractions 1) 0.077777777.... Convert 0.77777 to a fraction: Multiply by 10. (7.7777) Subtract, making sure to keep both sides balanced. 10 N = 7.7777... N = 0.7777... ---------------- 9 N = 7 Divide both sides by 9 and 0.777777 as a fraction is 7/9. Because the repeating numbers start in the hundredths place (second number to the right of the decimal, the denominator must be 2 digits. Therefore, you add a 0 to the end. (7/90) Answer: 7/90 2) 0.6777777 0.6 + 0.0777777 From before: 0.07777 as a fraction is 7/90 so 6/10 + 7/90 54/90 + 7/90 = 61/90 Date: 06/30/2008 at 13:36:00 From: Doctor Achilles Subject: Re: Converting repeating decimals into fractions Hi Sarah, Great work, you did those perfectly! Now, can you solve the problems you wrote in with? 0.91666666666666... 0.38888888888888... 0.84733333333333... If you're stuck on any of those or if you want me to double check an answer, please write back. As an extra challenge, you can try these out if you want: 0.7676767676767676... 55.7676767676767676... 0.557676767676767676... 0.12341234123412341234... Let me know what you get. If you're stuck and you want help with any of those, or if you have any other questions about repeating decimals, please let me know. Best wishes. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ Date: 07/01/2008 at 00:16:34 From: Sarah Subject: Converting repeating decimals into fractions Can you please make sure that these problems are all correct, especially the challenge problems. Thanks. 1) 0.91666666666666... 0.91 + 0.0066666 Convert 0.66666 to a fraction and get 6/9 or 2/3. Repeating numbers start in the thousandths place (third number right of the decimal), so denominator must be three digits so 2/300. 91/100 + 2/300 273/300 + 2/300 = 275/300 Simplify 275/300 = 11/12 Answer: 11/12 2) 0.38888888888888... 0.3 + 0.08888 Convert 0.8888 to a fraction and get 8/9. Repeating numbers start in the hundredths place (second number right of the decimal), so denominator must be two digits so 8/90. 3/10 + 8/90 27/90 + 8/90 = 35/90 Simplify 35/90 = 7/18 Answer: 7/18 3) 0.84733333333333... 0.847 + 0.0003333 Convert 0.3333 to a fraction and get 3/9 or 1/3. Repeating numbers start in the ten-thousandths place (fourth number right of the decimal), so denominator must be four digits so 1/3000. 847/1000 + 1/3000 2541/3000 + 1/3000 = 2542/3000 Simplify 2542/3000 = 1271/1500 Answer: 1271/1500 Extra Challenge: 1) 0.7676767676767676... Multiply by 10 to the power of two (100) (76.76767676767676) Subtract, making sure to keep both sides balanced. 100 N = 76.767676... N = 0.767676... -------------------- 99 N = 76 Divide both sides by 99. Answer: 76/99 2) 55.7676767676767676... 55 + 0.767676... As shown above, 0.767676... as a fraction is 76/99 Answer: 55 76/99 3) 0.557676767676767676... 0.55 + 0.0076767676 As shown above, 0.767676 as a fraction is 76/99 Repeating numbers start in the thousandths place, so denominator must add two digits so 76/9900. 55/100 + 76/9900 5445/9900 + 76/9900 = 5521/9900 Answer: 5521/9900 4) 0.12341234123412341234... Multiply by 10 to the power of 4. (10,000) 1234.123412341234 Subtract making sure to keep both sides balanced. 10,000 N = 1234.12341234... N = 0.12341234... --------------------------- 9,999 N = 1234 Divide both sides by 9,999 Answer: 1234/9999 Date: 07/01/2008 at 11:55:01 From: Doctor Achilles Subject: Re: Converting repeating decimals into fractions Hi Sarah, Excellent work, it looks like you've got the hang of this. You've done all the problems including the challenges perfectly. If you have any other questions, please let me know. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ Date: 07/01/2008 at 11:55:01 From: Doctor Riz Subject: Re: Converting repeating decimals into fractions Hi Sarah - Nice work with Dr. Achilles! I just wanted to show you a similar but slightly different way to do these problems, which builds in your step of changing the denominator depending on how many places there are before the repeating portion of the decimal starts. You've shown that you understand the idea of rewriting the number by multiplying by powers of 10, and that's the key to this approach. Let's look at 0.677777... again. My goal is to produce two versions of that number that both have the repeating 7 starting right after the decimal. Clearly one way to do that is to multiply by 10: N = 0.67777... 10 N = 6.77777... Now let's multiply the original number again. Since one number is repeating, we need one more power of 10 this time, or 100: N = 0.6777... 100 N = 67.7777... Now we can subtract the two equations that we made: 100 N = 67.7777... 10 N = 6.7777... ------------------ 90 N = 61 Dividing by 90, N = 61/90 Can you see how this method ensures the proper denominator without having to count the places and adjust the denominator as you did above? The key is to get the repeating parts to match up after the decimal. So, for a number like 0.12787878... we'd multiply by 10^2 or 100 to get the 78 to start repeating right after the decimal. Then, since two numbers are repeating, we'd multiply again by two more powers of 10, so 10^4 or 10,000: N = 0.127878... 10,000 N = 1278.7878... 100 N = 12.7878... ---------------------------- 9,900 N = 1266 Dividing by 9,900, we have N = 1266/9900, which reduces to 211/1650. You can decide which method you prefer. I just wanted to include this second approach for your consideration. Again, great work on the problems you did for Dr. Achilles! - Doctor Riz, The Math Forum http://mathforum.org/dr.math/ |
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