Converting 3.24 Repeating Decimal To A Fraction A Comprehensive Guide

Hey guys! Ever stumbled upon a number like 3.24 where that little line above the 4 (the vinculum) means it goes on forever – 44444…? These are called repeating decimals, and they might seem a bit tricky at first. But don't worry, we're going to break down how to rewrite this type of number into a good ol' fraction. This is super useful in math because fractions are often easier to work with than decimals, especially when you're doing things like algebra or calculus. So, let's dive into the world of repeating decimals and conquer this conversion challenge!

Understanding Repeating Decimals: The Key to Conversion

Before we jump into the nitty-gritty of converting 3.24, let's make sure we're all on the same page about what repeating decimals actually are. A repeating decimal, as the name suggests, is a decimal number where one or more digits repeat infinitely. The repeating part is indicated by that line (vinculum) drawn above the digit(s). So, in our case, 3.24 means 3.244444…, where the 4s go on forever.

Why do these repeating decimals even exist? Well, they often pop up when we try to express fractions as decimals. For example, if you divide 1 by 3, you get 0.3333…, which is a repeating decimal. Some fractions will result in terminating decimals (like 1/4 = 0.25), but others, like 1/3 or 1/6, will give you repeating decimals. Understanding this connection between fractions and repeating decimals is key to converting them back and forth. The conversion process relies on algebraic manipulation to eliminate the repeating part, leaving us with a whole number numerator and denominator. This might sound intimidating, but we'll walk through it step by step, and you'll see it's totally manageable. We'll use a clever trick involving multiplying the decimal by powers of 10 to shift the decimal point and then subtracting to get rid of the repeating part. This method works for any repeating decimal, whether it's a single digit repeating or a longer sequence of digits.

Step-by-Step Guide: Converting 3.24 to a Fraction

Okay, let's get down to business and convert 3.24 into a fraction. Here's the step-by-step process:

Step 1: Set up an equation.

Let's call our repeating decimal x. So, we have:

x = 3.24

This is our starting point. We're simply assigning the value of the repeating decimal to a variable, which will make our algebraic manipulations easier.

Step 2: Multiply by 10 to shift the decimal.

Since only one digit (the 4) is repeating, we'll multiply both sides of the equation by 10. This will shift the decimal point one place to the right:

10x = 32.4

Notice how the repeating part is still there after the decimal point. This is crucial for the next step.

Step 3: Multiply by 100 to shift the decimal again.

Now, we'll multiply both sides of the original equation by 100. This will shift the decimal point two places to the right:

100x = 324.4

We now have two equations: 10x = 32.4 and 100x = 324.4. The key is that both of these equations have the same repeating decimal part (.4444…). This is what we're going to exploit in the next step.

Step 4: Subtract the equations.

This is the magic step where we get rid of the repeating decimal! We'll subtract the first equation (10x = 32.4) from the second equation (100x = 324.4):

100x = 324.4

• 10x = 32.4

90x = 292

See how the repeating decimals cancelled each other out? This is because we carefully chose our multiples of 10 to ensure the repeating parts lined up perfectly. We're now left with a simple equation: 90x = 292.

Step 5: Solve for x.

To isolate x, we'll divide both sides of the equation by 90:

x = 292 / 90

Step 6: Simplify the fraction.

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

x = 146 / 45

And there you have it! We've successfully converted the repeating decimal 3.24 into the fraction 146/45.

Alternative Method: A Quick Trick for Repeating Decimals

Okay, guys, there's also a neat little trick you can use to convert repeating decimals, especially if you're in a hurry or just want a quicker way to check your work. This method is based on the same principles as the step-by-step method, but it condenses the process into a formula. Here's how it works:

1. Write down the repeating decimal without the decimal point and the bar. In our case, 3.24 becomes 324.
2. Subtract the non-repeating part. The non-repeating part is the digits before the repeating block, which is 32 in our case. So, we subtract 32 from 324: 324 - 32 = 292.
3. Write the result as the numerator of the fraction. So, our numerator is 292.
4. For the denominator, write as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the decimal point. In our case, there's one repeating digit (4), so we write a 9. There's also one non-repeating digit after the decimal point (2), so we add a 0. This gives us a denominator of 90.
5. Put it all together: The fraction is 292/90.
6. Simplify the fraction: Just like before, we simplify 292/90 to 146/45.

See? We got the same answer using this shortcut! This trick is a great way to quickly convert repeating decimals, but it's important to understand the underlying principles (the step-by-step method) so you know why it works.

Practice Makes Perfect: Examples and Exercises

Now that we've conquered 3.24, let's try a few more examples to really solidify your understanding. Remember, the key is to identify the repeating part, set up your equations, and use subtraction to eliminate the repeating decimal.

Example 1: Convert 0.7 to a fraction.

• Let x = 0.7
• 10x = 7.7
• Subtract the equations: 10x - x = 7.7 - 0.7, which gives us 9x = 7
• Solve for x: x = 7/9

Example 2: Convert 1.23 to a fraction.

• Let x = 1.23
• 100x = 123.23
• Subtract the equations: 100x - x = 123.23 - 1.23, which gives us 99x = 122
• Solve for x: x = 122/99

Exercises for you to try:

1. Convert 0.5 to a fraction.
2. Convert 2.16 to a fraction.
3. Convert 0.123 to a fraction.

Work through these exercises, and if you get stuck, review the steps we discussed earlier. Don't be afraid to use the quick trick to check your answers, but make sure you understand the step-by-step method as well.

Why This Matters: Real-World Applications and Further Exploration

You might be wondering,