Converting Rational Numbers To Fractions A Step By Step Guide

Hey guys! Today, we're diving into the fascinating world of rational numbers and how to express them as fractions. It's a fundamental concept in mathematics, and mastering it will open doors to more complex topics. So, let's get started and explore how to convert these decimals into their fractional forms. We'll tackle each example step-by-step, making it super easy to follow along. You'll be a pro in no time!

Understanding Rational Numbers

Before we jump into the conversions, let's quickly recap what rational numbers are. In essence, a rational number is any number that can be expressed as a fraction ${ \frac{p}{q} }$, where ${ p }$ and ${ q }$ are integers, and ${ q }$ is not equal to zero. This means that any number that can be written as a ratio of two whole numbers is considered rational. This category includes integers, fractions, terminating decimals, and repeating decimals. Think of it this way: if you can write it as a fraction, it's rational! Understanding this basic principle is crucial as we move forward.

Decimals, like the ones we're working with today, are a common way to represent rational numbers. But sometimes, we need to see them in their fractional form to perform certain operations or to better understand their value. The process of converting decimals to fractions involves recognizing the place value of the decimal digits and using that information to construct the fraction. It might seem a bit tricky at first, but once you get the hang of it, it's a piece of cake! We'll break down each step, so you'll see how straightforward it can be. Let's get started with our first example!

Converting Decimals to Fractions: Step-by-Step

Now, let's dive into the process of converting decimals to fractions. The key is to understand the place value of the digits after the decimal point. For example, the first digit after the decimal represents tenths, the second represents hundredths, the third represents thousandths, and so on. We'll use this knowledge to create our fractions. The general approach involves these steps:

1. Write down the decimal number.
2. Count the number of digits after the decimal point. This will determine the power of 10 we'll use as our denominator.
3. Write the number without the decimal point as the numerator.
4. Write 1 followed by the same number of zeros as the number of digits after the decimal point as the denominator.
5. Simplify the fraction if possible by dividing both the numerator and the denominator by their greatest common divisor (GCD).

A. 1.75

Let's start with 1.75. To convert this decimal to a fraction, we follow these steps:

1. The decimal number is 1.75.
2. There are two digits after the decimal point (7 and 5).
3. Write the number without the decimal point: 175.
4. Write the denominator as 1 followed by two zeros: 100.
5. So, we have the fraction ${ \frac{175}{100} }$. Now, we need to simplify this fraction. Both 175 and 100 are divisible by 25. Dividing both the numerator and the denominator by 25, we get ${ \frac{175 ÷ 25}{100 ÷ 25} = \frac{7}{4} }$. Therefore, 1.75 as a fraction is ${ \frac{7}{4} }$. You see, it's all about breaking it down into manageable steps!

B. 2.83

Next up, we have 2.83. Let's convert this decimal into its fractional form:

1. The decimal number is 2.83.
2. There are two digits after the decimal point (8 and 3).
3. Write the number without the decimal point: 283.
4. Write the denominator as 1 followed by two zeros: 100.
5. So, we have the fraction ${ \frac{283}{100} }$. Now, we check if we can simplify this fraction. 283 is a prime number, and it doesn't share any common factors with 100 other than 1. Therefore, the fraction is already in its simplest form. Thus, 2.83 as a fraction is ${ \frac{283}{100} }$. Sometimes, the fraction is already simplified, making our job easier!

C. 4.25

Now, let's tackle 4.25. We'll follow the same steps as before:

1. The decimal number is 4.25.
2. There are two digits after the decimal point (2 and 5).
3. Write the number without the decimal point: 425.
4. Write the denominator as 1 followed by two zeros: 100.
5. So, we have the fraction ${ \frac{425}{100} }$. To simplify, we can divide both the numerator and the denominator by their greatest common divisor, which is 25. Dividing both by 25, we get ${ \frac{425 ÷ 25}{100 ÷ 25} = \frac{17}{4} }$. So, 4.25 as a fraction is ${ \frac{17}{4} }$. Simplifying fractions is a crucial step to get the answer in its most concise form.

D. 10.48

Moving on to 10.48, let's convert this decimal to a fraction:

1. The decimal number is 10.48.
2. There are two digits after the decimal point (4 and 8).
3. Write the number without the decimal point: 1048.
4. Write the denominator as 1 followed by two zeros: 100.
5. So, we have the fraction ${ \frac{1048}{100} }$. To simplify, we can start by dividing both the numerator and the denominator by 2. This gives us ${ \frac{524}{50} }$. We can divide by 2 again to get ${ \frac{262}{25} }$. Since 262 and 25 don't have any common factors other than 1, the fraction is now in its simplest form. Therefore, 10.48 as a fraction is ${ \frac{262}{25} }$. Simplifying step-by-step can make the process less daunting.

E. 15.5

Let's convert 15.5 into a fraction. This one is slightly different as there's only one digit after the decimal point:

1. The decimal number is 15.5.
2. There is one digit after the decimal point (5).
3. Write the number without the decimal point: 155.
4. Write the denominator as 1 followed by one zero: 10.
5. So, we have the fraction ${ \frac{155}{10} }$. Both 155 and 10 are divisible by 5. Dividing both by 5, we get ${ \frac{155 ÷ 5}{10 ÷ 5} = \frac{31}{2} }$. Thus, 15.5 as a fraction is ${ \frac{31}{2} }$. Remember, the number of digits after the decimal point directly influences the denominator.

F. 23.9

Finally, let's convert 23.9 to a fraction. This one is similar to the previous example:

1. The decimal number is 23.9.
2. There is one digit after the decimal point (9).
3. Write the number without the decimal point: 239.
4. Write the denominator as 1 followed by one zero: 10.
5. So, we have the fraction ${ \frac{239}{10} }$. 239 is a prime number, and it doesn't share any common factors with 10 other than 1. Therefore, the fraction is already in its simplest form. Thus, 23.9 as a fraction is ${ \frac{239}{10} }$. Sometimes, you'll find that the fraction is already in its simplest form, saving you an extra step!

Conclusion

Alright, guys! We've successfully converted several rational numbers from their decimal form to fractions. Remember, the key is to understand the place value of the decimal digits and use that to construct your fraction. Don't forget to simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor. Converting decimals to fractions is a fundamental skill in mathematics, and with practice, you'll become more confident and proficient. So, keep practicing, and you'll master it in no time! If you ever get stuck, just revisit these steps, and you'll be on the right track. Happy converting!