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What is Mixed Recurring Decimal?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A mixed recurring decimal is a decimal number where some digits after the decimal point do NOT repeat, but then one or more digits start repeating in a pattern forever. It's like a mix of a non-repeating part and a repeating part.
Simple Example
Quick Example
Imagine you are converting 1/6 into a decimal. You get 0.1666... Here, '1' appears just once after the decimal, but '6' keeps repeating. This '0.1666...' is a mixed recurring decimal because '1' is the non-repeating part and '6' is the repeating part.
Worked Example
Step-by-Step
Let's convert the fraction 5/12 into a decimal to see if it's a mixed recurring decimal.
Step 1: Divide 5 by 12.
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Step 2: 5 divided by 12 is 0 with a remainder of 5. Add a decimal and a zero: 50.
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Step 3: 50 divided by 12 is 4 with a remainder of 2. (12 x 4 = 48).
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Step 4: Add another zero: 20. 20 divided by 12 is 1 with a remainder of 8. (12 x 1 = 12).
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Step 5: Add another zero: 80. 80 divided by 12 is 6 with a remainder of 8. (12 x 6 = 72).
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Step 6: Add another zero: 80. 80 divided by 12 is 6 with a remainder of 8. (12 x 6 = 72).
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Step 7: We can see that '6' will keep repeating. So, 5/12 = 0.41666...
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Answer: 0.41666... is a mixed recurring decimal, where '41' is the non-repeating part and '6' is the repeating part.
Why It Matters
Understanding these decimals helps scientists and engineers in fields like data science and physics to work with precise numbers that don't end. For example, in computer science, these concepts are crucial for accurate calculations in programming and cryptography, which keeps your online data safe.
Common Mistakes
MISTAKE: Thinking that any repeating decimal is a mixed recurring decimal. For example, 0.333... | CORRECTION: 0.333... is a pure recurring decimal because the repetition starts immediately after the decimal point. A mixed recurring decimal must have some non-repeating digits first.
MISTAKE: Confusing the non-repeating part with the repeating part. For example, in 0.2555..., thinking '25' is the repeating part. | CORRECTION: In 0.2555..., only '5' is repeating. '2' is the non-repeating part, and '5' is the repeating part. The bar is placed only over the repeating digits.
MISTAKE: Writing only a few repeating digits and thinking it's exact. For example, writing 0.166 for 1/6. | CORRECTION: Mixed recurring decimals go on forever. To show this, use an ellipsis (...) or a bar over the repeating digit(s). For 1/6, write 0.166... or 0.1(bar over 6).
Practice Questions
Try It Yourself
QUESTION: Is 0.727272... a mixed recurring decimal? | ANSWER: No, it is a pure recurring decimal because the repetition (72) starts immediately after the decimal point.
QUESTION: Identify the non-repeating and repeating parts in 0.12333... | ANSWER: Non-repeating part: 12. Repeating part: 3.
QUESTION: Convert the fraction 7/15 into a decimal. Is it a mixed recurring decimal? If yes, identify the non-repeating and repeating parts. | ANSWER: 7 divided by 15 is 0.4666... Yes, it is a mixed recurring decimal. Non-repeating part: 4. Repeating part: 6.
MCQ
Quick Quiz
Which of the following is a mixed recurring decimal?
0.3333...
0.121212...
0.5444...
0.75
The Correct Answer Is:
C
Option C (0.5444...) has '5' as a non-repeating digit followed by '4' which repeats, making it a mixed recurring decimal. Options A and B are pure recurring, and Option D is a terminating decimal.
Real World Connection
In the Real World
Mixed recurring decimals pop up when you're calculating things that involve division that doesn't terminate easily. For instance, when calculating batting averages in cricket or fuel efficiency for vehicles, sometimes the exact decimal value might be a mixed recurring decimal. Data scientists analyzing large datasets also encounter these numbers when dealing with proportions or ratios.
Key Vocabulary
Key Terms
DECIMAL: A number that includes a decimal point, separating the whole number part from the fractional part. | RECURRING DECIMAL: A decimal where one or more digits repeat infinitely after the decimal point. | PURE RECURRING DECIMAL: A recurring decimal where the repeating part starts immediately after the decimal point. | NON-REPEATING PART: The digits after the decimal point that do not repeat in a mixed recurring decimal. | REPEATING PART: The digit(s) that repeat infinitely in a recurring decimal.
What's Next
What to Learn Next
Great job understanding mixed recurring decimals! Next, you can learn how to convert these mixed recurring decimals back into fractions (p/q form). This will help you see the connection between fractions and these special decimals even more clearly.
