What Is The Fraction Of 1.2

Converting a decimal like 1.2 into a fraction involves understanding the relationship between decimal places and fractions, and then simplifying the resulting fraction to its lowest terms. This process is fundamental in mathematics and has practical applications in various fields, from cooking and construction to finance and engineering.

Understanding Decimals and Fractions

A decimal is a number expressed in the base-10 system, using a decimal point to separate the whole number part from the fractional part. Each digit to the right of the decimal point represents a power of 10, such as tenths, hundredths, thousandths, and so on.

A fraction, on the other hand, represents a part of a whole and is expressed as a ratio between two numbers: the numerator (the top number) and the denominator (the bottom number). The denominator indicates how many equal parts the whole is divided into, and the numerator indicates how many of those parts are being considered.

The key to converting a decimal to a fraction is to recognize the place value of the decimal digits and express them as fractions with powers of 10 as denominators.

Converting 1.2 to a Fraction: Step-by-Step

Converting 1.2 to a fraction is a straightforward process. Here’s a detailed, step-by-step guide:

Step 1: Identify the Decimal Place Value

The decimal 1.2 has one digit after the decimal point, which means it extends to the tenths place. This is a crucial piece of information because it determines the denominator of the fraction.

Step 2: Write the Decimal as a Fraction

To convert 1.2 to a fraction, you can write it as "1 and 2 tenths." In fractional form, this is written as:

1 + 2/10

Here, "1" represents the whole number part, and "2/10" represents the decimal part as a fraction.

Step 3: Convert the Whole Number to a Fraction

To combine the whole number and the fraction, you need to convert the whole number to a fraction with the same denominator as the fractional part. In this case, the denominator is 10. So, you convert "1" to a fraction with a denominator of 10:

1 = 10/10

Step 4: Combine the Fractions

Now, you can add the two fractions together:

10/10 + 2/10 = (10 + 2) / 10 = 12/10

So, 1.2 is equal to 12/10 as a fraction.

Step 5: Simplify the Fraction

The fraction 12/10 is not in its simplest form. To simplify it, you need to find the greatest common divisor (GCD) of the numerator (12) and the denominator (10) and divide both numbers by the GCD.

The factors of 12 are: 1, 2, 3, 4, 6, and 12. The factors of 10 are: 1, 2, 5, and 10.

The greatest common divisor of 12 and 10 is 2. Now, divide both the numerator and the denominator by 2:

12 ÷ 2 = 6
10 ÷ 2 = 5

So, the simplified fraction is 6/5.

Step 6: Express as a Mixed Number (Optional)

The fraction 6/5 is an improper fraction because the numerator is greater than the denominator. You can convert it to a mixed number, which is a whole number and a proper fraction combined.

To convert 6/5 to a mixed number, divide 6 by 5:

6 ÷ 5 = 1 with a remainder of 1

This means that 6/5 is equal to 1 whole and 1/5. Therefore, the mixed number is:

1 1/5

So, 1.2 can be expressed as the fraction 6/5 or the mixed number 1 1/5.

Practical Examples

Here are a few practical examples of how converting decimals to fractions can be useful:

Cooking

In cooking, recipes often call for measurements in fractions, but sometimes you might have measurements in decimals. For example, a recipe might call for 1.2 cups of flour. To measure this accurately using standard measuring cups, you would convert 1.2 to 1 1/5 cups.

Construction

In construction and woodworking, precise measurements are essential. If a blueprint specifies a length of 1.2 inches, you might need to convert this to a fraction to use a ruler or measuring tape. Converting 1.2 to 6/5 inches allows for more accurate measurements.

Finance

In finance, understanding fractions and decimals is crucial for calculating interest rates, returns on investments, and other financial metrics. Converting decimals to fractions can help in comparing different financial options or understanding the composition of a financial product.

Common Mistakes to Avoid

When converting decimals to fractions, there are several common mistakes to avoid:

Forgetting to Simplify

One of the most common mistakes is forgetting to simplify the fraction to its lowest terms. Always ensure that the numerator and denominator have no common factors other than 1.

Misidentifying the Decimal Place Value

Incorrectly identifying the decimal place value can lead to an incorrect fraction. For example, confusing tenths with hundredths will result in a wrong denominator.

Incorrectly Converting Whole Numbers

When dealing with decimals that have a whole number part, make sure to correctly convert the whole number to a fraction with the same denominator as the decimal part.

Arithmetic Errors

Simple arithmetic errors during the conversion and simplification process can lead to incorrect results. Double-check all calculations to ensure accuracy.

The Mathematics Behind Converting Decimals to Fractions

The process of converting decimals to fractions is rooted in the principles of place value and the base-10 number system. Each decimal place represents a negative power of 10. For example:

The first digit after the decimal point represents tenths (10^-1 = 1/10)
The second digit represents hundredths (10^-2 = 1/100)
The third digit represents thousandths (10^-3 = 1/1000)

When converting a decimal to a fraction, you are essentially expressing the decimal as a sum of fractions with powers of 10 as denominators. For example, the decimal 0.75 can be expressed as:

7/10 + 5/100 = 70/100 + 5/100 = 75/100

This fraction can then be simplified to its lowest terms, which in this case is 3/4.

Similarly, when converting a decimal with a whole number part, you are combining the whole number with the fractional part. For example, the decimal 2.45 can be expressed as:

2 + 4/10 + 5/100 = 200/100 + 40/100 + 5/100 = 245/100

This fraction can then be simplified to its lowest terms, which in this case is 49/20.

Advanced Concepts

Repeating Decimals

Converting repeating decimals to fractions requires a different approach. A repeating decimal is a decimal in which one or more digits repeat infinitely. For example, 0.333... is a repeating decimal.

To convert a repeating decimal to a fraction, you can use algebraic methods. Here’s how to convert 0.333... to a fraction:

Let x = 0.333... Multiply both sides by 10: 10x = 3.333... Subtract the first equation from the second equation: 10x - x = 3.333... - 0.333... 9x = 3 Divide both sides by 9: x = 3/9 Simplify the fraction: x = 1/3

So, 0.333... is equal to 1/3 as a fraction.

Non-Repeating Decimals

Non-repeating decimals that do not terminate (e.g., π ≈ 3.141592653589793...) cannot be expressed as exact fractions. These are irrational numbers. However, they can be approximated to a certain number of decimal places and then converted to a fraction.

Why Is This Important?

Understanding how to convert decimals to fractions is essential for several reasons:

Mathematical Proficiency

It enhances your mathematical skills and understanding of the relationship between different types of numbers.

Problem Solving

It allows you to solve problems in various fields that require converting between decimals and fractions.

Precision

It helps in achieving greater precision in measurements and calculations, especially in fields like engineering, construction, and finance.

Standardized Tests

It is a fundamental concept that is often tested in standardized math exams.

Frequently Asked Questions (FAQ)

What is the fraction of 1.2?

The fraction of 1.2 is 6/5, which can also be expressed as the mixed number 1 1/5.

How do you convert a decimal to a fraction?

To convert a decimal to a fraction:

Identify the decimal place value.
Write the decimal as a fraction with a denominator that is a power of 10.
Simplify the fraction to its lowest terms.

What is a mixed number?

A mixed number is a combination of a whole number and a proper fraction. For example, 1 1/5 is a mixed number.

How do you convert an improper fraction to a mixed number?

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part, and the remainder is the numerator of the fractional part. The denominator remains the same.

Can all decimals be converted to fractions?

No, not all decimals can be converted to exact fractions. Repeating decimals can be converted to fractions using algebraic methods, but non-repeating, non-terminating decimals (irrational numbers) cannot be expressed as exact fractions.

What is the greatest common divisor (GCD)?

The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. It is used to simplify fractions to their lowest terms.

Why is it important to simplify fractions?

Simplifying fractions makes them easier to understand and compare. It also ensures that the fraction is in its most concise form.

How do you find the GCD of two numbers?

There are several methods to find the GCD of two numbers, including:

Listing the factors of each number and finding the largest common factor.
Using the Euclidean algorithm, which involves repeatedly dividing the larger number by the smaller number until the remainder is 0. The GCD is the last non-zero remainder.

What is the difference between a rational and an irrational number?

A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to 0. An irrational number is a number that cannot be expressed as a fraction and has a non-repeating, non-terminating decimal representation.

Conclusion

Converting the decimal 1.2 to a fraction involves understanding the place value of decimals, writing the decimal as a fraction, and simplifying the fraction to its lowest terms. The fraction of 1.2 is 6/5, which can also be expressed as the mixed number 1 1/5. This skill is essential for various applications in mathematics, science, and everyday life. By mastering the process of converting decimals to fractions, you can enhance your mathematical proficiency and problem-solving abilities.