How to Write Repeating Decimals as Fractions
Have you ever found yourself staring at a repeating decimal, like 0.777… or 0.2̅, and wondered how on earth it could be expressed as a fraction? You’re not alone! Many people encounter this conundrum in math class or while tackling everyday calculations. But fear not; converting these seemingly complex numbers into fractions is simpler than it appears.
Let’s dive into the process with an example that many of us can relate to: the repeating decimal 0.777…. To start, we’ll assign a variable to our decimal:
Step 1: Set Up Your Equation
Let ( x = 0.777…). This equation represents our repeating decimal.
Step 2: Create Another Equation
Next, multiply both sides of your original equation by 10 (this shifts the decimal point one place to the right):
[10x = 7.777…
]
Now we have two equations:
- ( x = 0.777…)
- (10x = 7.777…)
Step 3: Subtract One Equation from the Other
The beauty of this method lies in its simplicity—by subtracting the first equation from the second, we eliminate those pesky repeating decimals:
[10x – x = 7.777… – 0.777…
] This simplifies down to: [
9x = 7
]
Step 4: Solve for ( x )
Now it’s time for some straightforward algebra! Divide both sides by nine:
[x = \frac{7}{9}
]
And there you have it! The repeating decimal (0.\overline{7}) is equivalent to the fraction (\frac{7}{9}).
But what if you’re faced with another type of repeating decimal? Let’s explore another common scenario using (0.\overline{2}), which means that "2" repeats indefinitely.
Converting Another Example
Again, let’s set up our variable:
-
Define Your Variable:
Let ( y = 0.\overline{2} ). -
Multiply by Ten:
[
Multiply both sides by ten:
y = .222… \
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Subtracting gives us:
```markdown
y(10 - y) =
Which leads us back through similar steps until ultimately arriving at:
y =
So once again we’ve transformed a simple concept into something tangible!
Why It Matters
Understanding how to convert repeating decimals into fractions isn’t just about passing tests; it’s about grasping fundamental mathematical concepts that will serve you well throughout life—whether you’re budgeting your finances or trying out new recipes where precision matters.
Next time you come across a number like 3. followed closely behind by endless zeros or any other digits circling endlessly around infinity’s edge don’t panic instead remember these easy steps take control over those infinite sequences turn them back into manageable forms and enjoy newfound confidence when facing mathematics head-on!
In conclusion writing out long division may seem tedious but practice makes perfect so keep working on examples until they become second nature soon enough you’ll find yourself breezing through problems effortlessly transforming decimals all around without breaking sweat.