Question

Evaluate the expression. \(\frac{1}{2}+\left(-\frac{4}{5}\right)-\frac{2}{3}\)

Short answer

The solution to the expression \(\frac{1}{2}+\left(-\frac{4}{5}\right)-\frac{2}{3}\) is \(-\frac{29}{30}\).

Step-by-step solution

Understand the Negative Fraction

In the given expression, there is a fraction \(-\frac{4}{5}\). The negative sign in front of this fraction indicates that this fraction should be subtracted from other terms of the expression.

Add/Subtract Fractions

To add or subtract fractions, first find a common denominator. The denominators of the fractions are 2, 5, and 3. Their least common multiplier is 30. So, convert all fractions with the denominator of 30, then add and subtract them: \(\frac{1}{2}=\frac{15}{30}\), \(-\frac{4}{5}=-\frac{24}{30}\), and \(-\frac{2}{3}=-\frac{20}{30}\). After converting, the expression becomes: \(\frac{15}{30}-\frac{24}{30}-\frac{20}{30}\).

Sum up the fractions

Now, simply sum up the fractions: \(\frac{15-24-20}{30}=\frac{-29}{30}\)

Key concepts

Adding and Subtracting Fractions

When it comes to dealing with fractions, the process of addition and subtraction can initially seem challenging, but with a clear understanding of the principles, it becomes much more straightforward. To successfully add or subtract fractions, you first need to ensure the fractions have a common denominator. This is essential because fractions represent parts of a whole, and to combine or compare these parts, you need to refer to the same size of the whole; in other words, identical denominators.

Consider a simple analogy: if you wanted to combine two groups of slices from different-sized pizzas, comparing them wouldn't make much sense until you cut all slices to match in size, symbolizing a common denominator. Once this step is completed, you can simply add or subtract the numerators, which are the top numbers of the fractions. Here's a quick rundown on how you would handle this process numerically:

• Identify the denominators of the fractions you are working with.
• Find a common denominator, preferably the least common denominator (LCD), to avoid unnecessary simplification later.
• Convert each fraction to an equivalent fraction with the newly found common denominator.
• Add or subtract the numerators of these converted fractions.
• If needed, simplify or reduce the resulting fraction to its simplest form.

Negative Fractions

Negative fractions can sometimes confuse students, but they adhere to the same rules as positive fractions. A negative fraction simply means that the quantity represented by the fraction is being subtracted, or it may signify a direction opposite to positive fractions.

The most important thing to remember about negative fractions is that the negative sign applies to the entire fraction, not just the numerator or denominator. In mathematical operations, such as the given exercise, you handle negative fractions by including the negative sign when adding or subtracting. It's crucial to pay attention to the signs when combining them with other fractions. Here's a straightforward approach:

• Keep track of the negative sign: treat it as if it's part of the numerator.
• When adding a negative fraction, you're essentially subtracting its absolute value.
• When subtracting a negative fraction, you're technically adding its absolute value, due to the rule that subtracting a negative is the same as adding a positive.

Learning the behavior of negative fractions enables you to maneuver through expressions that include a mix of positive and negative values with confidence.

Least Common Denominator

The idea of the least common denominator (LCD) is central to combining fractions through addition or subtraction. The LCD is the smallest number that all denominators can divide into without any remainder. It is also known as the least common multiple of the denominators. Finding the LCD reduces the need for further simplification after adding or subtracting fractions and ensures that the fractions are being compared or combined on a uniform scale.

Finding the LCD may seem demanding, but one efficient method is to list the multiples of the largest denominator and then identify the smallest multiple that the other denominators can divide into. Alternatively, you can use the prime factorization method, which involves breaking down each denominator into its prime factors and then using the highest powers of these primes to form the LCD.

In practice, these steps can help identify the LCD:

• List the multiples of each denominator until you find a common multiple.
• Or, use prime factorization of the denominators to identify the LCD.
• Convert all fractions to equivalent fractions with the LCD to perform addition or subtraction.

Understanding how to find and use the LCD allows for efficient and accurate manipulation of fractions, as was necessary in the original exercise presented.