Problem 60 - Prealgebra

Question

Simplify the given expression by first converting the decimal into a fraction. \(-\frac{5}{6}+2.3\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{22}{15}\).

Step 1: Convert the Decimal to a Fraction

First, we need to convert the decimal number 2.3 into a fraction. The decimal 2.3 can be written as \(2 + 0.3\). The decimal 0.3 is equal to \(\frac{3}{10}\). Therefore, \(2.3\) can be written as the mixed number \(2\frac{3}{10}\) or as the improper fraction \(\frac{23}{10}\) (2 times 10 plus 3).

Step 2: Find a Common Denominator

To simplify the expression, we need to add the fractions \(-\frac{5}{6}\) and \(\frac{23}{10}\). First, find a common denominator for these fractions. The least common multiple of 6 and 10 is 30. Therefore, we need to rewrite each fraction with a denominator of 30.

Step 3: Rewrite the Fractions with the Common Denominator

For \(-\frac{5}{6}\), multiply the numerator and the denominator by 5 to obtain \(-\frac{25}{30}\). For \(\frac{23}{10}\), multiply the numerator and the denominator by 3 to obtain \(\frac{69}{30}\). Now both fractions have a common denominator of 30.

Step 4: Add the Fractions

Now that both fractions have the same denominator, add them: \(-\frac{25}{30} + \frac{69}{30} = \frac{-25 + 69}{30} = \frac{44}{30}\).

Step 5: Simplify the Result

Simplify \(\frac{44}{30}\) by finding the greatest common divisor of 44 and 30. The greatest common divisor is 2. Divide both the numerator and the denominator by 2: \(\frac{44}{30} = \frac{22}{15}\).

Key Concepts

Decimal to Fraction Conversion

Converting a decimal into a fraction can seem tricky at first, but it becomes simple with a systematic approach. A decimal represents a part of a whole number, which correlates directly to fractions. For example, to convert the decimal number 2.3 into a fraction, we first separate it into 2 and 0.3. The decimal 0.3 can be translated into \(\frac{3}{10}\). Thus, 2.3 is expressed as the mixed number \(2\frac{3}{10}\).
For further simplification, transform this mixed number into an improper fraction. Multiply 2 by 10 and add 3, resulting in \(\frac{23}{10}\). This conversion is an essential skill in mathematical manipulations as it allows for smooth operations like addition and subtraction with other fractions.

Least Common Multiple

The least common multiple (LCM) is fundamental when working with fractions needing a common denominator. To add or subtract fractions, they must share the same denominator. Let's explore this concept using our problem.
Consider fractions \(\frac{-5}{6}\) and \(\frac{23}{10}\). Find the LCM of 6 and 10 to establish a common denominator. List multiples of each:

• Multiples of 6: 6, 12, 18, 24, 30...
• Multiples of 10: 10, 20, 30...

The smallest shared multiple is 30, hence the LCM is 30.
Using the LCM simplifies addition/subtraction by creating a common base for calculations. It ensures precision and clarity in solving fraction-based problems efficiently.

Adding Fractions

Adding fractions can seem complex due to varying denominators, but the process becomes seamless when converted to a common denominator. Consider \(\frac{-5}{6}\) and \(\frac{23}{10}\), with our previously found common denominator of 30.
Each fraction must be adjusted to this new common denominator. Multiply the numerator and denominator of \(\frac{-5}{6}\) by 5, resulting in \(\frac{-25}{30}\). Next, convert \(\frac{23}{10}\) by multiplying by 3, yielding \(\frac{69}{30}\).
With equivalent denominators, add the fractions: \(\frac{-25}{30} + \frac{69}{30} = \frac{44}{30}\). This approach neatly aligns the fractions for accurate addition, enabling easier calculations.

Simplifying Fractions

Once fractions are added or subtracted, the resulting fraction often needs simplification. This entails reducing the fraction to its smallest possible form for clarity and conciseness.
To simplify \(\frac{44}{30}\) from our example, discover the greatest common divisor (GCD) between 44 and 30. Here, the GCD is 2. Divide both the numerator and denominator by this number:

• 44 divided by 2 equals 22
• 30 divided by 2 equals 15

Thus, \(\frac{44}{30}\) simplifies to \(\frac{22}{15}\). Simplification aids in recognizing and understanding the fraction in its most basic form, enhancing readability and usability of the final result.