Decoding Decimals: Turning 0.63 Repeating into a Fraction – It’s Easier Than You Think (Probably)
Have you looked at 0.636363… and felt dread? It seems endless. This repeating decimal may seem abstract. But here’s the twist: it’s a normal fraction. Yes, infinite decimals can become neat fractions. Let’s figure out how to convert 0.63 repeating into a fraction. It’s simpler than you think.
Understanding the Repeating Decimal Drama
First, what is a repeating decimal? It’s a decimal where one or more digits repeat endlessly. Our example, 0.636363…, shows “63” repeating. This repetition helps us convert it into a fraction. Think of it like a catchy song stuck in your head. We will change the tune to a fraction.
The Algebraic Method: Unleashing the Math Magic
Now, let’s convert 0.63 repeating into a fraction. We’ll use a bit of algebra. Set our repeating decimal as ‘x’.
So, let x = 0.636363…
The repeating part is “63”. It has two digits. To shift the decimal point two places, multiply both sides by 100. Why 100? Because it matches the count of repeating digits. If it was 0.6 repeating, we’d multiply by 10. If it was 0.635 repeating, we’d multiply by 1000.
100x = 63.636363…
Now comes the clever part. We subtract our original equation from this new one:
100x = 63.636363…
x = 0.636363…
——————–
99x = 63
Notice how the repeating decimals cancel! We have a simple equation: 99x = 63. To find x, divide both sides by 99:
x = 63 / 99
And that’s it! 0.63 repeating equals the fraction 63/99. You can find more details at MathTutor. They explain it with enthusiasm.
Simplifying the Fraction: Because Neatness Counts
While 63/99 is correct, it’s not simplified. Fractions are better in their simplest form. To simplify 63/99, we need the greatest common factor (GCF) of 63 and 99. The GCF is the largest number dividing evenly into both.
Factors of 63: 1, 3, 7, 9, 21, 63.
Factors of 99: 1, 3, 9, 11, 33, 99.
The greatest common factor is 9. Now, divide both the numerator (63) and denominator (99) by 9:
63 ÷ 9 = 7
99 ÷ 9 = 11
The simplified fraction is 7/11. Thus, 0.63 repeating in simplest form is 7/11. Congratulations! You turned a repeating decimal into a fraction.
General Rule for Repeating Decimals: A Quick Shortcut
Want a shortcut for repeating decimals? There’s a handy general rule. For a repeating decimal where all digits repeat (like 0.7 repeating), it’s just the repeat digit(s) over 9s. For example, for 0.7 repeating, it’s 7/9. For 0.63 repeating, it’s 63/99, which we simplify. For 1.2 repeating, separate the whole number first. 1.2 repeating is 1 + 0.2 repeating. The latter is 2/9, so combine to get 1 + 2/9, which is 11/9. This rule works well, but the algebraic method is reliable for all repeating decimals.
Exploring Other Repeating Decimal Adventures
Let’s look at more examples to solidify our understanding and impress friends.
- 0.6 repeating (0.666…): Using our rule, it’s simply 6/9, which simplifies to 2/3.
- 0.8 repeating (0.8̅): Following the rule again, this is 8/9. No simplification needed.
- 0.72 repeating (0.72̅): If ’72’ repeats like this (0.727272…), it’s 72/99. Simplifying gives us 8/11.
- 1/3 as a repeating decimal: Divide 1 by 3 to get 0.333…, or 0.3 repeating.
Decimals to Percentages: A Quick Detour
Now let’s convert decimals to percentages briefly. It’s straightforward: multiply any decimal by 100. So, for example, 0.63 as a percentage is 0.63 * 100 = 63%. Easy! This is useful to while we focus on fractions.
Terminating Decimals to Fractions: The Simpler Cousins
Let’s differentiate between repeating and terminating decimals. Terminating decimals end; they don’t go on forever, like 0.63, 0.063, or 0.062. Converting these to fractions is simple.
- 0.63: With two decimal places, it becomes 63 over 100: 63/100.
- 0.063: Three decimal places? Now it’s 63 over 1000: 63/1000.
- 0.062: Three decimal places give us 62 over 1000. This one can simplify! Both divide by 2: 62 ÷ 2 = 31 and 1000 ÷ 2 = 500. Thus, it’s simplified form is 31/500.
See the difference? Terminating decimals become fractions with powers of ten for denominators, depending on decimal places. Repeating decimals use nines in the denominator after our algebraic dance.
Calculator to the Rescue (Sometimes)
Finally, for simplifying fractions, calculators can help! Many have fraction simplification functions. If you’re ever uncertain about a fraction’s simplest form or want to double-check it, a calculator is handy. However, learning by hand earns you mathematical credit.
So that’s it! Converting 0.63 repeating to a fraction isn’t daunting as it seems. With algebra and some simplification skills, you can transform repeating decimals into fractions confidently. Math can be manageable! If you need more explanations or examples, websites like Brainly can provide insights. Now go forth and conquer those decimals!
