What Is 1.75 As A Fraction

Converting decimals to fractions is a fundamental skill in mathematics, and understanding how to perform this conversion for a number like 1.75 is crucial for various practical applications. Whether you're working on a math problem, adjusting measurements in a recipe, or dealing with financial calculations, the ability to seamlessly switch between decimals and fractions is invaluable. This article delves into the process of converting 1.75 into a fraction, explaining each step in detail to ensure a clear and comprehensive understanding.

Understanding Decimals and Fractions

Before we dive into the conversion process, it's essential to understand the basics of decimals and fractions.

• Decimals: Decimals are a way of representing numbers that are not whole. They use a base-10 system, where each digit to the right of the decimal point represents a fraction with a denominator of 10, 100, 1000, and so on. For example, in the decimal 1.75, the '7' represents 7 tenths (7/10), and the '5' represents 5 hundredths (5/100).
• Fractions: Fractions represent parts of a whole. They consist of two main components: the numerator (the top number) and the denominator (the bottom number). The numerator indicates how many parts of the whole are being considered, while the denominator indicates the total number of equal parts that make up the whole. For example, in the fraction 3/4, '3' is the numerator, and '4' is the denominator, indicating that we are considering 3 parts out of a total of 4 equal parts.

The Conversion Process: Step-by-Step Guide

Converting the decimal 1.75 into a fraction involves a few straightforward steps. Here’s a detailed guide to help you through the process:

Step 1: Identify the Decimal

The decimal we want to convert is 1.75. This number has a whole number part (1) and a decimal part (.75).

Step 2: Express the Decimal as a Fraction over a Power of 10

To convert 1.75 to a fraction, we need to express it as a fraction with a denominator that is a power of 10 (10, 100, 1000, etc.). The number of decimal places determines the power of 10 we use. Since 1.75 has two decimal places, we will use 100 as the denominator.

So, we can write 1.75 as 175/100. This is because 1.75 is the same as saying "175 hundredths."

Step 3: Simplify the Fraction

The fraction 175/100 can be simplified by finding the greatest common divisor (GCD) of the numerator (175) and the denominator (100) and then dividing both by the GCD.

To find the GCD of 175 and 100, we can use the Euclidean algorithm or simply list the factors of each number:

• Factors of 175: 1, 5, 7, 25, 35, 175
• Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

The greatest common divisor of 175 and 100 is 25.

Now, divide both the numerator and the denominator by 25:

• 175 ÷ 25 = 7
• 100 ÷ 25 = 4

So, the simplified fraction is 7/4.

Step 4: Express as a Mixed Number (Optional)

The fraction 7/4 is an improper fraction because the numerator (7) is greater than the denominator (4). We can convert this improper fraction into a mixed number, which consists of a whole number and a proper fraction.

To convert 7/4 to a mixed number, divide 7 by 4:

• 7 ÷ 4 = 1 with a remainder of 3

The quotient (1) is the whole number part of the mixed number, and the remainder (3) becomes the numerator of the fractional part, with the original denominator (4) remaining the same.

So, 7/4 as a mixed number is 1 3/4.

Alternative Method: Breaking Down the Decimal

Another way to approach this conversion is by breaking down the decimal into its whole number and fractional parts separately and then combining them.

Step 1: Separate the Whole Number and Decimal Parts

In the decimal 1.75, the whole number part is 1, and the decimal part is 0.75.

Step 2: Convert the Decimal Part to a Fraction

Convert 0.75 to a fraction. Since there are two decimal places, we can write 0.75 as 75/100.

Step 3: Simplify the Fraction

Simplify the fraction 75/100 by finding the GCD of 75 and 100, which is 25.

• 75 ÷ 25 = 3
• 100 ÷ 25 = 4

So, the simplified fraction is 3/4.

Step 4: Combine the Whole Number and the Fraction

Now, combine the whole number (1) and the fraction (3/4) to form a mixed number: 1 3/4.

Step 5: Convert to an Improper Fraction (Optional)

If you need an improper fraction, convert the mixed number 1 3/4 back to an improper fraction:

• Multiply the whole number (1) by the denominator (4): 1 × 4 = 4
• Add the numerator (3): 4 + 3 = 7
• Place the result over the original denominator (4): 7/4

So, the improper fraction is 7/4.

Practical Applications

Understanding how to convert decimals to fractions is useful in various real-world scenarios:

• Cooking and Baking: Recipes often use fractions to specify ingredient amounts. If you need to scale a recipe that uses decimals, converting to fractions can make it easier to measure ingredients accurately.
• Construction and Measurement: In construction, measurements are often given in decimals or fractions. Converting between the two can help ensure precision in cutting materials and assembling structures.
• Financial Calculations: When dealing with money, decimals and fractions are frequently used. Converting between them can simplify calculations and make it easier to understand financial transactions.
• Academic Math: Mastering the conversion between decimals and fractions is essential for success in algebra, geometry, and other advanced math courses.

Common Mistakes to Avoid

When converting decimals to fractions, there are a few common mistakes to watch out for:

• Incorrectly Identifying the Decimal Places: Make sure you correctly count the number of decimal places, as this determines the power of 10 you use as the denominator. For example, 0.75 has two decimal places, so the denominator should be 100, not 10 or 1000.
• Forgetting to Simplify the Fraction: Always simplify the fraction to its lowest terms. This makes the fraction easier to work with and ensures that your answer is in the simplest form.
• Errors in Finding the Greatest Common Divisor (GCD): Ensure you accurately find the GCD of the numerator and denominator. A mistake in finding the GCD will result in an incorrect simplified fraction.
• Improper Conversion to Mixed Numbers: When converting an improper fraction to a mixed number, make sure you correctly divide the numerator by the denominator and accurately represent the quotient and remainder.

Examples and Practice Problems

To solidify your understanding, let's work through a few examples:

Example 1: Convert 2.25 to a Fraction

1. Identify the Decimal: 2.25
2. Express as a Fraction: 225/100
3. Simplify the Fraction:
   • GCD of 225 and 100 is 25.
   • 225 ÷ 25 = 9
   • 100 ÷ 25 = 4
   • Simplified fraction: 9/4
4. Express as a Mixed Number (Optional):
   • 9 ÷ 4 = 2 with a remainder of 1
   • Mixed number: 2 1/4

Example 2: Convert 0.6 to a Fraction

1. Identify the Decimal: 0.6
2. Express as a Fraction: 6/10
3. Simplify the Fraction:
   • GCD of 6 and 10 is 2.
   • 6 ÷ 2 = 3
   • 10 ÷ 2 = 5
   • Simplified fraction: 3/5

Practice Problems:

1. Convert 3.5 to a fraction.
2. Convert 1.125 to a fraction.
3. Convert 0.8 to a fraction.
4. Convert 2.75 to a fraction.
5. Convert 0.333 to a fraction (approximate).

Advanced Tips and Tricks

• Recognizing Common Decimal-Fraction Equivalents: Familiarize yourself with common decimal-fraction equivalents like 0.25 = 1/4, 0.5 = 1/2, 0.75 = 3/4, and 0.125 = 1/8. This can speed up the conversion process.
• Using Prime Factorization: If you struggle to find the GCD, use prime factorization. Break down the numerator and denominator into their prime factors and cancel out the common factors.
• Approximating Repeating Decimals: For repeating decimals like 0.333..., recognize that they can be represented as fractions (in this case, 1/3). Some repeating decimals may require algebraic methods to convert accurately.

The Scientific Explanation

The conversion from decimals to fractions is rooted in the base-10 number system. Decimals are a way of representing numbers in terms of powers of 10. Each digit to the right of the decimal point represents a fraction with a denominator that is a power of 10.

For example, the decimal 1.75 can be expressed as:

1 + (7 × 10^-1) + (5 × 10^-2) = 1 + (7/10) + (5/100)

To convert this to a single fraction, we find a common denominator, which in this case is 100:

(100/100) + (70/100) + (5/100) = 175/100

Simplifying this fraction involves finding the greatest common divisor (GCD), which is based on the fundamental theorem of arithmetic. This theorem states that every integer greater than 1 can be uniquely represented as a product of prime numbers. By finding the prime factorization of both the numerator and the denominator, we can identify and cancel out common factors, leading to the simplest form of the fraction.

FAQs

Q: Why is it important to simplify fractions after converting from decimals?

A: Simplifying fractions provides the most concise and easily understandable representation of the number. It also ensures that the fraction is in its lowest terms, making it easier to work with in further calculations.

Q: Can all decimals be converted into fractions?

A: Yes, all terminating decimals (decimals that end) and repeating decimals can be converted into fractions. Non-repeating, non-terminating decimals (irrational numbers) cannot be expressed as exact fractions but can be approximated.

Q: What is a mixed number, and how does it relate to improper fractions?

A: A mixed number is a combination of a whole number and a proper fraction (where the numerator is less than the denominator). Improper fractions (where the numerator is greater than or equal to the denominator) can be converted into mixed numbers to provide a more intuitive representation of the quantity.

Q: How do I convert a decimal with more than two decimal places to a fraction?

A: The process is the same. For example, to convert 1.125 to a fraction, express it as 1125/1000 and then simplify by finding the GCD of 1125 and 1000 (which is 125). So, 1125/1000 simplifies to 9/8.

Q: What if I have a repeating decimal like 0.333...?

A: Repeating decimals can be converted to fractions using algebraic methods. For example, let x = 0.333... Then 10x = 3.333... Subtracting x from 10x gives 9x = 3, so x = 3/9, which simplifies to 1/3.

Conclusion

Converting the decimal 1.75 to a fraction involves expressing it as 175/100 and then simplifying to 7/4, which can also be represented as the mixed number 1 3/4. This process is essential for various practical applications, from cooking and construction to financial calculations and academic math. By understanding the steps involved and avoiding common mistakes, you can confidently convert decimals to fractions and enhance your mathematical skills. Practice regularly with different examples to solidify your understanding and build fluency in this important skill.