How To Convert The Following Decimal Numbers To Ordinary Fractions: a) 2.47 b) 0.38 c) 1.2 d) 0.(3) e) 1.(7) f) 2.(23) g) 1.11(3) h) 2.0(4) i) 12.1(13)

Jun 20, 2025 by ADMIN 152 views

Converting Decimal Numbers to Fractions: A Comprehensive Guide

Decimal numbers are a common part of our daily lives, appearing in everything from prices and measurements to scientific calculations. However, understanding how to convert these decimal numbers into ordinary fractions is a fundamental skill in mathematics. This article provides a comprehensive guide on how to transform various types of decimal numbers into their equivalent fractional forms. We will cover terminating decimals, repeating decimals, and mixed repeating decimals, providing clear explanations and examples for each type. By the end of this guide, you will have a solid understanding of how to convert any decimal number into a fraction, enhancing your mathematical toolkit and problem-solving abilities.

a) 2.47

To convert the decimal number 2.47 into an ordinary fraction, we first need to recognize that 2.47 is a terminating decimal. Terminating decimals are decimal numbers that have a finite number of digits after the decimal point. The process of converting such decimals into fractions involves a few straightforward steps. First, we write the decimal number as a fraction with a denominator that is a power of 10. The power of 10 is determined by the number of digits after the decimal point. In the case of 2.47, there are two digits after the decimal point, so we will use 10 squared, which is 100, as our denominator. This gives us 247/100.

Now, we write 2.47 as a fraction: 247/100. This fraction represents the decimal number, but to ensure we have the simplest form, we need to check if the fraction can be simplified further. To simplify a fraction, we look for common factors between the numerator (247) and the denominator (100). A common factor is a number that divides both the numerator and the denominator without leaving a remainder. In this case, the prime factors of 247 are 13 and 19, while the prime factors of 100 are 2 and 5. Since there are no common factors between 247 and 100, the fraction 247/100 is already in its simplest form. Therefore, the decimal number 2.47 is equivalent to the ordinary fraction 247/100. This method works because we are essentially expressing the decimal as a sum of whole numbers and fractions, and then combining them into a single fraction.

b) 0.38

Converting 0.38 to an ordinary fraction follows a similar process to the previous example, as 0.38 is also a terminating decimal. The key here is to understand that the digits after the decimal point represent fractional parts of the whole number. In this case, 0.38 means 38 hundredths. To convert 0.38 into a fraction, we first write it as a fraction with a denominator of 100 because there are two digits after the decimal point. This gives us 38/100. Now, we need to simplify this fraction to its lowest terms. Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD. The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

For the fraction 38/100, we need to find the GCD of 38 and 100. The factors of 38 are 1, 2, 19, and 38, while the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The greatest common factor between 38 and 100 is 2. Therefore, we divide both the numerator and the denominator by 2: (38 ÷ 2) / (100 ÷ 2) = 19/50. The fraction 19/50 is now in its simplest form because 19 is a prime number and does not share any common factors with 50 other than 1. Thus, the decimal 0.38 is equivalent to the fraction 19/50. This process illustrates how terminating decimals can be easily converted into fractions by expressing them as hundredths, thousandths, or other powers of ten, and then simplifying the resulting fraction.

c) 1.2

The conversion of the decimal number 1.2 into an ordinary fraction is another example of dealing with a terminating decimal. The process involves recognizing the place value of the digits after the decimal point. In this case, 1.2 has one digit after the decimal point, which represents tenths. To convert 1.2 into a fraction, we first write it as a mixed number or an improper fraction. A mixed number combines a whole number and a fraction, while an improper fraction has a numerator greater than its denominator. We can write 1.2 as 1 and 2/10. Now, we convert the fractional part, 2/10, to its simplest form. The factors of 2 are 1 and 2, and the factors of 10 are 1, 2, 5, and 10. The greatest common factor between 2 and 10 is 2. Dividing both the numerator and the denominator by 2, we get (2 ÷ 2) / (10 ÷ 2) = 1/5. So, 1.2 can be written as 1 and 1/5.

To further simplify, we can convert the mixed number 1 and 1/5 into an improper fraction. To do this, we multiply the whole number (1) by the denominator (5) and add the numerator (1), then place the result over the original denominator. This gives us (1 × 5 + 1) / 5 = 6/5. Therefore, the decimal number 1.2 is equivalent to the ordinary fraction 6/5. Alternatively, we can directly write 1.2 as 12/10 and then simplify. The GCD of 12 and 10 is 2. Dividing both the numerator and the denominator by 2, we get (12 ÷ 2) / (10 ÷ 2) = 6/5. This demonstrates another method to convert terminating decimals into fractions, highlighting the importance of simplifying fractions to their lowest terms for clarity and consistency in mathematical expressions.

d) 0.(3)

The decimal number 0.(3) represents a repeating decimal, which means the digit 3 repeats infinitely. Converting repeating decimals into fractions requires a slightly different approach compared to terminating decimals. To convert 0.(3) into a fraction, we can use an algebraic method. Let x = 0.(3). This means x = 0.3333... Now, we multiply both sides of the equation by 10 because the repeating part has one digit. This gives us 10x = 3.3333...

Next, we subtract the original equation (x = 0.3333...) from the new equation (10x = 3.3333...). This step is crucial because it eliminates the repeating part of the decimal. Subtracting the equations, we get 10x - x = 3.3333... - 0.3333..., which simplifies to 9x = 3. Now, we solve for x by dividing both sides of the equation by 9: x = 3/9. To simplify the fraction, we find the greatest common divisor (GCD) of 3 and 9, which is 3. Dividing both the numerator and the denominator by 3, we get (3 ÷ 3) / (9 ÷ 3) = 1/3. Therefore, the repeating decimal 0.(3) is equivalent to the ordinary fraction 1/3. This method elegantly uses algebra to eliminate the repeating part of the decimal, allowing us to express it as a precise fraction.

e) 1.(7)

To convert the repeating decimal 1.(7) into an ordinary fraction, we again use the algebraic method similar to the previous example. Here, the digit 7 repeats infinitely. Let x = 1.(7). This means x = 1.7777... To eliminate the repeating part, we multiply both sides of the equation by 10, since there is one repeating digit. This gives us 10x = 17.7777...

Now, we subtract the original equation (x = 1.7777...) from the new equation (10x = 17.7777...). Subtracting the equations, we get 10x - x = 17.7777... - 1.7777..., which simplifies to 9x = 16. To solve for x, we divide both sides of the equation by 9: x = 16/9. The fraction 16/9 is already in its simplest form because 16 and 9 do not have any common factors other than 1. Thus, the repeating decimal 1.(7) is equivalent to the ordinary fraction 16/9. This fraction can also be expressed as a mixed number, which is 1 and 7/9. The process illustrates how multiplying by a power of 10 and subtracting the original equation effectively cancels out the repeating part, leaving a simple equation to solve for the fractional equivalent.

f) 2.(23)

Converting the repeating decimal 2.(23) into an ordinary fraction involves a slightly more complex application of the algebraic method, as two digits (23) repeat infinitely. Let x = 2.(23). This means x = 2.232323... Since two digits repeat, we multiply both sides of the equation by 100 to shift the repeating block to the left of the decimal point. This gives us 100x = 223.232323...

Next, we subtract the original equation (x = 2.232323...) from the new equation (100x = 223.232323...). This subtraction eliminates the repeating decimal part. The equation becomes 100x - x = 223.232323... - 2.232323..., which simplifies to 99x = 221. Now, we solve for x by dividing both sides of the equation by 99: x = 221/99. To simplify the fraction, we look for common factors between 221 and 99. The prime factors of 221 are 13 and 17, while the prime factors of 99 are 3 and 11. Since there are no common factors between 221 and 99, the fraction 221/99 is already in its simplest form. Therefore, the repeating decimal 2.(23) is equivalent to the ordinary fraction 221/99. This fraction can also be expressed as a mixed number, which is 2 and 23/99. This method demonstrates the importance of multiplying by the appropriate power of 10 (100 in this case) to align the repeating blocks for subtraction.

g) 1.11(3)

The decimal number 1.11(3) is a mixed repeating decimal, which means it has a non-repeating part (1.11) and a repeating part (3). Converting mixed repeating decimals into fractions requires a combination of the methods used for terminating and repeating decimals. Let x = 1.11(3). This means x = 1.113333... First, we multiply both sides of the equation by 100 to move the non-repeating part to the left of the decimal point. This gives us 100x = 111.3333...

Next, we need to address the repeating part. We multiply both sides of the new equation by 10 to shift one repeating block to the left. This gives us 1000x = 1113.3333... Now, we subtract the equation 100x = 111.3333... from 1000x = 1113.3333... to eliminate the repeating part. Subtracting the equations, we get 1000x - 100x = 1113.3333... - 111.3333..., which simplifies to 900x = 1002. To solve for x, we divide both sides of the equation by 900: x = 1002/900. To simplify the fraction, we find the greatest common divisor (GCD) of 1002 and 900. The GCD of 1002 and 900 is 6. Dividing both the numerator and the denominator by 6, we get (1002 ÷ 6) / (900 ÷ 6) = 167/150. Therefore, the mixed repeating decimal 1.11(3) is equivalent to the ordinary fraction 167/150. This method demonstrates the systematic steps required to handle mixed repeating decimals, ensuring accurate conversion to fractional form.

h) 2.0(4)

To convert the mixed repeating decimal 2.0(4) into an ordinary fraction, we follow a similar procedure as in the previous example. The decimal number 2.0(4) means 2.04444..., where the digit 4 repeats infinitely. Let x = 2.0(4). We first multiply both sides of the equation by 10 to move the non-repeating part (0) to the left of the repeating digit. This gives us 10x = 20.4444...

Now, we multiply both sides of the equation by 10 again to shift one repeating block to the left. This results in 100x = 204.4444... Next, we subtract the equation 10x = 20.4444... from 100x = 204.4444... to eliminate the repeating part. This gives us 100x - 10x = 204.4444... - 20.4444..., which simplifies to 90x = 184. To solve for x, we divide both sides of the equation by 90: x = 184/90. To simplify the fraction, we find the greatest common divisor (GCD) of 184 and 90. The GCD of 184 and 90 is 2. Dividing both the numerator and the denominator by 2, we get (184 ÷ 2) / (90 ÷ 2) = 92/45. Therefore, the mixed repeating decimal 2.0(4) is equivalent to the ordinary fraction 92/45. This fraction can also be expressed as a mixed number, which is 2 and 2/45. The systematic application of algebraic techniques allows for the accurate conversion of mixed repeating decimals into their fractional equivalents.

i) 12.1(13)

Converting the mixed repeating decimal 12.1(13) into an ordinary fraction requires a detailed application of the algebraic method. The decimal number 12.1(13) means 12.1131313..., where the digits 13 repeat infinitely after the non-repeating digit 1. Let x = 12.1(13). First, we multiply both sides of the equation by 10 to move the non-repeating part to the left of the repeating block. This gives us 10x = 121.131313...

Next, since two digits repeat, we multiply both sides of the new equation by 100 to shift one repeating block to the left. This results in 1000x = 12113.131313... Now, we subtract the equation 10x = 121.131313... from 1000x = 12113.131313... to eliminate the repeating part. This gives us 1000x - 10x = 12113.131313... - 121.131313..., which simplifies to 990x = 11992. To solve for x, we divide both sides of the equation by 990: x = 11992/990. To simplify the fraction, we find the greatest common divisor (GCD) of 11992 and 990. The GCD of 11992 and 990 is 2. Dividing both the numerator and the denominator by 2, we get (11992 ÷ 2) / (990 ÷ 2) = 5996/495. We can further simplify this fraction by finding the GCD of 5996 and 495, which is 1. Thus, the fraction 5996/495 is in its simplest form. Therefore, the mixed repeating decimal 12.1(13) is equivalent to the ordinary fraction 5996/495. This complex example underscores the importance of precise algebraic manipulation and simplification to accurately convert mixed repeating decimals into fractions.

Conclusion

In conclusion, converting decimal numbers into ordinary fractions is a crucial skill in mathematics, with applications ranging from basic arithmetic to advanced calculations. This guide has covered the methods for converting terminating decimals, repeating decimals, and mixed repeating decimals into their equivalent fractional forms. For terminating decimals, the process involves writing the decimal as a fraction with a power of 10 as the denominator and then simplifying. Repeating decimals require an algebraic approach, where multiplying by a power of 10 and subtracting the original equation eliminates the repeating part. Mixed repeating decimals combine aspects of both methods, requiring careful manipulation to isolate and eliminate the repeating portion.

By mastering these techniques, you gain a deeper understanding of the relationship between decimals and fractions, enhancing your ability to solve a wide range of mathematical problems. The examples provided illustrate the step-by-step procedures, allowing for clear and accurate conversions. Whether you are a student learning these concepts for the first time or someone looking to refresh your skills, this guide provides a comprehensive resource for transforming decimal numbers into ordinary fractions. The ability to confidently perform these conversions not only strengthens your mathematical foundation but also enhances your problem-solving toolkit in various real-world scenarios.