Problem 61 - Prealgebra

Question

In Exercises 61-64, simplify the given expression. \(\frac{5}{6}+2.375\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{77}{24} \).

Step 1: Identify the Components

The task involves adding a fraction \( \frac{5}{6} \) and a decimal number 2.375.

Step 2: Convert Decimal to Fraction

Convert the decimal 2.375 into a fraction. Recognize that 2.375 equals 2 + 0.375. The fraction equivalent of 0.375 is \( \frac{375}{1000} \), which can be simplified by dividing both the numerator and the denominator by 125, resulting in \( \frac{3}{8} \). This gives us \( 2 + \frac{3}{8} \).

Step 3: Combine the Mixed Fraction

Combine the whole number 2 with the fraction \( \frac{3}{8} \) to form \( \frac{16}{8} + \frac{3}{8} = \frac{19}{8} \). This makes 2.375 equivalent to \( \frac{19}{8} \).

Step 4: Find a Common Denominator

Determine the least common denominator (LCD) for the fractions \( \frac{5}{6} \) and \( \frac{19}{8} \). The LCD of 6 and 8 is 24.

Step 5: Convert Fractions to Common Denominator

Convert \( \frac{5}{6} \) to the common denominator: \( \frac{5}{6} = \frac{20}{24} \). Convert \( \frac{19}{8} \) to the common denominator: \( \frac{19}{8} = \frac{57}{24} \).

Step 6: Add the Fractions

Now, add the fractions \( \frac{20}{24} \) and \( \frac{57}{24} \) together:\( \frac{20}{24} + \frac{57}{24} = \frac{77}{24} \).

Step 7: Simplify the Result

The fraction \( \frac{77}{24} \) is already in its simplest form, and thus, this is the simplified expression.

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole, allowing us to express parts of a whole or a division of quantities. Essentially, a fraction consists of a numerator and a denominator, with the numerator indicating the number of parts we have, and the denominator showing the total number of equal parts that make up the whole.

For example, the fraction \( \frac{5}{6} \) tells us we have 5 parts out of a total of 6 equal parts. Understanding fractions requires familiarity with some key concepts:

• Proper fractions: The numerator is less than the denominator (e.g., \( \frac{5}{6} \)).
• Improper fractions: The numerator is greater than or equal to the denominator (e.g., \( \frac{19}{8} \)). These can also be written as mixed numbers, such as \( 2 + \frac{3}{8} \).
• Simplifying fractions: Reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor.

Working with fractions is essential, especially when combining them with other numbers, as seen in the simplification process of our exercise.

Decimal Conversion

Converting decimals to fractions is a fundamental skill when simplifying expressions involving both types of numbers. A decimal, such as 2.375, represents a fraction whose denominator is a power of ten, because it is based on the base-ten number system.

To convert decimals to fractions, follow these steps:

• Identify the place value: 2.375 can be split into a whole number (2) and a decimal part (0.375), where 0.375 is in the thousandths place because it has three digits after the decimal point.
• Write as a fraction: 0.375 becomes \( \frac{375}{1000} \) because the '375' reaches the thousandths place.
• Simplify the fraction: Divide the numerator and the denominator by the greatest common divisor (125 in this case) to obtain \( \frac{3}{8} \).

After conversion, combine the whole number with the fraction (\( \frac{16}{8} \) to \( \frac{19}{8} \)) to complete the transformation, as shown in our exercise solution.

Common Denominator

When adding or subtracting fractions, finding a common denominator is crucial. This means making the denominators the same, so the fractions can be easily combined. In our exercise, we needed to find a common denominator for \( \frac{5}{6} \) and \( \frac{19}{8} \).

Here’s how to find a common denominator effectively:

• Determine the least common denominator (LCD): Identify the smallest number that each denominator can divide into evenly. For 6 and 8, 24 is the least common multiple.
• Adjust the fractions: Convert each fraction so the denominators are equal to the LCD. Multiply the numerator and denominator of \( \frac{5}{6} \) by 4 to get \( \frac{20}{24} \). Then, multiply both parts of \( \frac{19}{8} \) by 3 to get \( \frac{57}{24} \).

Once the fractions share a common denominator, adding them is straightforward: combine the numerators while keeping the denominator the same. In this case, the resulting \( \frac{77}{24} \) is already in its simplest form, which simplifies our expression perfectly.