7.75 As A Fraction In Simplest Form

7.75 as a Fraction in Simplest Form: A Comprehensive Guide

Converting decimals to fractions might seem daunting at first, but with a clear understanding of the process, it becomes surprisingly straightforward. This comprehensive guide will walk you through converting 7.75 into its simplest fractional form, explaining the steps involved and offering additional insights into working with decimals and fractions. We'll cover the fundamental concepts, demonstrate the solution, and provide you with extra practice problems to solidify your understanding.

Understanding Decimals and Fractions

Before diving into the conversion, let's refresh our understanding of decimals and fractions.

Decimals: Decimals represent numbers that are not whole numbers. They use a decimal point to separate the whole number part from the fractional part. For example, in 7.75, '7' is the whole number part, and '.75' is the fractional part. Each digit to the right of the decimal point represents a power of ten: tenths, hundredths, thousandths, and so on.

Fractions: Fractions represent parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). The numerator indicates how many parts you have, and the denominator indicates how many parts make up the whole. For instance, ½ represents one out of two equal parts.

Converting 7.75 to a Fraction: Step-by-Step

The key to converting a decimal to a fraction is to understand the place value of the last digit in the decimal.

Step 1: Identify the Place Value

In 7.75, the last digit, 5, is in the hundredths place. This means the decimal can be expressed as 7 and 75 hundredths.

Step 2: Write as an Improper Fraction

We can write 7.75 as a mixed number: 7 ⁷⁵⁄₁₀₀. To convert this mixed number into an improper fraction, we multiply the whole number (7) by the denominator (100) and add the numerator (75). This result becomes the new numerator, while the denominator remains the same.

7 x 100 + 75 = 775

Therefore, the improper fraction is ⁷⁷⁵⁄₁₀₀.

Step 3: Simplify the Fraction

The fraction ⁷⁷⁵⁄₁₀₀ is not in its simplest form. To simplify, we need to find the greatest common divisor (GCD) of the numerator (775) and the denominator (100). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the GCD can be done through several methods:

• Prime Factorization: Break down both numbers into their prime factors. The GCD is the product of the common prime factors raised to the lowest power.

  • 775 = 5² x 31
  • 100 = 2² x 5²
  • The common prime factor is 5², which is 25.

• Euclidean Algorithm: This algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD.

Let's use the Euclidean Algorithm:

1. Divide 775 by 100: 775 = 7 x 100 + 75
2. Divide 100 by 75: 100 = 1 x 75 + 25
3. Divide 75 by 25: 75 = 3 x 25 + 0

The last non-zero remainder is 25, so the GCD is 25.

Step 4: Divide the Numerator and Denominator by the GCD

Now, divide both the numerator and denominator of ⁷⁷⁵⁄₁₀₀ by the GCD (25):

⁷⁷⁵ ÷ 25 = 31 ₁₀₀ ÷ 25 = 4

This gives us the simplified fraction: ³¹⁄₄

Therefore, 7.75 as a fraction in its simplest form is ³¹⁄₄.

Further Understanding: Working with Decimals and Fractions

Mastering the conversion between decimals and fractions is a crucial skill in mathematics. Here are some additional points to enhance your understanding:

Converting Fractions to Decimals

The reverse process – converting a fraction to a decimal – is equally important. To convert a fraction to a decimal, simply divide the numerator by the denominator. For example:

³¹⁄₄ = 31 ÷ 4 = 7.75

This confirms our conversion from decimal to fraction.

Recurring Decimals

Not all decimals convert neatly into simple fractions. Some decimals are recurring, meaning they have a pattern of digits that repeat infinitely. For instance, ⅓ = 0.3333... These recurring decimals require a slightly different approach to convert them to fractions, often involving algebraic manipulation.

Mixed Numbers and Improper Fractions

Remember the distinction between mixed numbers (a whole number and a fraction) and improper fractions (where the numerator is larger than the denominator). Understanding how to convert between these forms is essential for simplifying fractions effectively.

Practice Problems

Let's test your understanding with some practice problems:

1. Convert 3.25 into a fraction in its simplest form.
2. Convert 0.625 into a fraction in its simplest form.
3. Convert 12.8 into a fraction in its simplest form.
4. Convert 2/5 into a decimal.
5. Convert 5/8 into a decimal.

Solutions:

1. ³¹⁄₄
2. ⁵⁄₈
3. ¹²⁸⁄₁₀ or ⁶⁴⁄₅
4. 0.4
5. 0.625

Conclusion

Converting decimals to fractions is a fundamental mathematical skill. By understanding the place value of decimals, applying the steps outlined above, and practicing regularly, you can confidently convert any decimal to its simplest fractional form. This skill is invaluable in various fields, from everyday calculations to more advanced mathematical applications. Remember to always simplify your fractions to their lowest terms for the most accurate and concise representation.