Convert Decimals To Rational Numbers Easily

by TextBrain Team 44 views

Hey guys! Let's break down how to turn those tricky decimals into neat, rational numbers. We'll tackle each example step by step, making sure everything is crystal clear. Get ready to sharpen those math skills!

a) Converting 0.24 to a Rational Number

When converting 0.24 to a rational number, the goal is to express it as a fraction in the form of p/q, where p and q are integers, and q is not zero. Here’s how we do it:

  1. Recognize that 0.24 is the same as 24 hundredths.
  2. Write it as a fraction: 24/100.
  3. Simplify the fraction by finding the greatest common divisor (GCD) of 24 and 100. The GCD is 4.
  4. Divide both the numerator and the denominator by 4:
    • 24 ÷ 4 = 6
    • 100 ÷ 4 = 25
  5. The simplified fraction is 6/25.

So, 0.24 as a rational number is 6/25. This method ensures that you convert the decimal accurately into its simplest fractional form, making it easier to work with in various mathematical operations.

Understanding this process is super helpful, especially when you're dealing with more complex numbers or need to perform calculations where fractions are easier to manage than decimals. Keep practicing, and you'll become a pro at converting decimals to rational numbers in no time!

b) Converting 1.23 to a Rational Number

Converting 1.23 to a rational number involves a similar approach to the previous example, but with an added whole number part. Here’s a detailed breakdown:

  1. Separate the whole number and the decimal part: 1 + 0.23.
  2. Convert the decimal part, 0.23, into a fraction. Recognize that 0.23 is 23 hundredths.
  3. Write it as a fraction: 23/100. This fraction is already in its simplest form because 23 is a prime number and does not share any common factors with 100 other than 1.
  4. Combine the whole number and the fraction: 1 + 23/100.
  5. To combine these, convert the whole number to a fraction with the same denominator as the decimal fraction:
    • 1 = 100/100
  6. Add the two fractions:
    • 100/100 + 23/100 = 123/100

Therefore, 1.23 as a rational number is 123/100. This fraction represents the decimal exactly in its rational form. This method is straightforward and ensures accuracy, especially when dealing with mixed numbers and decimals. Understanding how to break down the number into its components (whole number and decimal) makes the conversion process much easier and less prone to errors. Practice this method, and you’ll find it becomes second nature! Remember, the key is to convert the decimal portion into a fraction and then combine it with the whole number.

c) Converting 3.5 to a Rational Number

When you're looking at converting 3.5 to a rational number, remember we need to express it as a fraction. Here’s how to do it:

  1. Recognize that 3.5 is the same as 3 and 5 tenths.
  2. Write it as a mixed number: 3 1/2.
  3. Convert the mixed number to an improper fraction:
    • Multiply the whole number by the denominator: 3 * 2 = 6
    • Add the numerator: 6 + 1 = 7
    • Place the result over the original denominator: 7/2

So, 3.5 as a rational number is 7/2. Alternatively, you can think of 3.5 as 35 tenths (35/10), and then simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 5:

  • 35 ÷ 5 = 7
  • 10 ÷ 5 = 2

This gives you the same result: 7/2. This conversion is quite common and simple, making it a fundamental skill in dealing with rational numbers. Whether you choose to go through the mixed number route or directly convert to a fraction and simplify, the result will be the same. Keep practicing, and you'll master it in no time!

d) Converting 0.3 to a Rational Number

To convert 0.3 to a rational number, we aim to express it in the form of a fraction p/q. Here's the process:

  1. Recognize that 0.3 is the same as 3 tenths.
  2. Write it directly as a fraction: 3/10.

Since 3 is a prime number and does not share any common factors with 10 other than 1, the fraction 3/10 is already in its simplest form. Therefore, 0.3 as a rational number is simply 3/10. This is one of the most straightforward decimal-to-fraction conversions you can encounter. It’s direct and requires no further simplification, making it an easy concept to grasp. Keep it in mind as a quick win when you're practicing your conversions!

e) Converting 0.(2) to a Rational Number

Alright, let's tackle converting 0.(2) to a rational number. The notation 0.(2) means 0.2222..., which is a repeating decimal. Here's how to convert it:

  1. Let x = 0.2222...
  2. Multiply x by 10: 10x = 2.2222...
  3. Subtract the original equation from the new equation:
    • 10x - x = 2.2222... - 0.2222...
    • 9x = 2
  4. Solve for x:
    • x = 2/9

Thus, 0.(2) as a rational number is 2/9. This method works because the repeating part cancels out when you subtract the original number from the multiplied number. Understanding this technique is crucial for converting any repeating decimal into a fraction. Practice it a few times, and you'll be able to convert repeating decimals like a pro! The key is to set up the equations correctly and subtract to eliminate the repeating part.

f) Converting 0.(18) to a Rational Number

Now, let's see converting 0.(18) to a rational number. The notation 0.(18) means 0.181818..., a repeating decimal. Here's the conversion process:

  1. Let x = 0.181818...
  2. Multiply x by 100 (since the repeating part has two digits): 100x = 18.181818...
  3. Subtract the original equation from the new equation:
    • 100x - x = 18.181818... - 0.181818...
    • 99x = 18
  4. Solve for x:
    • x = 18/99
  5. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9:
    • 18 ÷ 9 = 2
    • 99 ÷ 9 = 11

So, 0.(18) as a rational number is 2/11. Remember, the key to this method is multiplying by a power of 10 that matches the length of the repeating part. This allows the repeating decimals to align and cancel out during subtraction. Keep practicing, and you’ll master converting repeating decimals to fractions in no time!

g) Converting -1.(13) to a Rational Number

Converting -1.(13) to a rational number is similar to the previous repeating decimal examples, but with an added negative sign and a whole number part. Here’s how we do it:

  1. Recognize that -1.(13) means -1.131313...
  2. Let x = 1.131313... (we'll handle the negative sign at the end).
  3. Multiply x by 100 (since the repeating part has two digits): 100x = 113.131313...
  4. Subtract the original equation from the new equation:
    • 100x - x = 113.131313... - 1.131313...
    • 99x = 112
  5. Solve for x:
    • x = 112/99
  6. Now, apply the negative sign: -112/99.

Thus, -1.(13) as a rational number is -112/99. This method ensures that you correctly handle both the repeating decimal and the negative sign. It's essential to keep track of the sign throughout the process to avoid errors. Remember, when dealing with negative numbers, treat them separately and combine the sign at the end.

h) Converting 0.2(45) to a Rational Number

Alright, let's tackle converting 0.2(45) to a rational number. This one's a bit trickier because only part of the decimal repeats. Here’s how to handle it:

  1. Let x = 0.2454545...
  2. Multiply x by 10 to move the non-repeating part to the left of the decimal point: 10x = 2.454545...
  3. Multiply 10x by 100 to move one repeating block to the left: 1000x = 245.454545...
  4. Subtract the second equation (10x) from the third equation (1000x):
    • 1000x - 10x = 245.454545... - 2.454545...
    • 990x = 243
  5. Solve for x:
    • x = 243/990
  6. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9:
    • 243 ÷ 9 = 27
    • 990 ÷ 9 = 110

Thus, 0.2(45) as a rational number is 27/110. This method works by strategically moving the decimal point to align the repeating parts for subtraction. It’s a bit more complex, but with practice, you'll get the hang of it! Remember to identify the non-repeating and repeating parts correctly to set up your equations accurately.

i) Converting -0.2(36) to a Rational Number

Lastly, let's convert -0.2(36) to a rational number. This is another mixed repeating decimal with a negative sign. Here’s the step-by-step process:

  1. Let x = 0.2363636... (ignore the negative sign for now, and add it back at the end).
  2. Multiply x by 10 to move the non-repeating part to the left of the decimal: 10x = 2.363636...
  3. Multiply 10x by 100 to move one repeating block to the left: 1000x = 236.363636...
  4. Subtract the second equation (10x) from the third equation (1000x):
    • 1000x - 10x = 236.363636... - 2.363636...
    • 990x = 234
  5. Solve for x:
    • x = 234/990
  6. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 18:
    • 234 ÷ 18 = 13
    • 990 ÷ 18 = 55
  7. Apply the negative sign: -13/55

Therefore, -0.2(36) as a rational number is -13/55. Remember to handle the negative sign separately and apply it at the end. This ensures you don't make sign errors during the conversion process. With a bit of practice, you’ll find these conversions become much easier. The key is to carefully set up your equations and subtract to eliminate the repeating parts.

Hope this helps you guys understand how to convert decimals to rational numbers! Keep practicing, and you'll become a math whiz in no time!