Problem 30 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$-2 \frac{3}{4}-1 \frac{1}{8}$$

Step-by-Step Solution

Answer

The sum is \(-\frac{31}{8}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(-2 \frac{3}{4}\), multiply the whole number 2 by the denominator 4 and add the numerator 3:\[ 2 \times 4 + 3 = 8 + 3 = 11. \]Thus, \(-2 \frac{3}{4}\) becomes \(-\frac{11}{4}\).Now for \(-1 \frac{1}{8}\), multiply the whole number 1 by the denominator 8 and add the numerator 1:\[ 1 \times 8 + 1 = 8 + 1 = 9. \]Thus, \(-1 \frac{1}{8}\) becomes \(-\frac{9}{8}\).

Step 2: Find Common Denominator

The denominators we have are 4 and 8. The least common denominator (LCD) is 8 since 8 is a multiple of 4.Convert \(-\frac{11}{4}\) to a fraction with a denominator of 8:\[ -\frac{11}{4} = -\frac{11 \times 2}{4 \times 2} = -\frac{22}{8}. \]

Step 3: Add the Fractions

Now add the two fractions: \(-\frac{22}{8}\) and \(-\frac{9}{8}\):\[ -\frac{22}{8} - \frac{9}{8} = -\frac{22 + 9}{8} = -\frac{31}{8}. \]

Step 4: Simplify the Fraction

Since \(-\frac{31}{8}\) is already in its simplest form, no further simplification is necessary.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top part) is greater than or equal to the denominator (the bottom part). This means the fraction represents a value greater than or equal to one whole. For example, the fraction \( \frac{11}{4} \) is improper, as 11 is greater than 4. When dealing with improper fractions, it's important to understand how they work because they make adding, subtracting, multiplying, or dividing easier.

• They can be converted to mixed numbers to better understand their value.
• They simplify calculations in equations and expressions involving fractions.

Improper fractions are often used in math because they provide a consistent format to work with. Converting mixed numbers to improper fractions, as shown in the exercise, helps unify the method for adding fractions. This way, all fractions share the same format, making calculations straightforward.

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are handy because they show a value as it is, instead of converting it all to parts of a fraction. For example, \( -2 \frac{3}{4} \) means "negative two and three-quarters," which is more intuitive than stating the value purely in terms of improper fractions.

• Mixed numbers are practical in everyday measurements like recipes or distances.
• They express a part-whole relationship, useful in visualization.

However, for arithmetic operations like addition or subtraction, mixed numbers often need to be converted to improper fractions. This conversion makes it easier to ensure consistency in mathematical operations, as seen in converting the mixed numbers in the exercise into improper fractions before proceeding with the calculations.

Common Denominator

In order to add or subtract fractions, they need to have the same denominator—the bottom part of the fraction. A common denominator allows the fractions to be directly added or subtracted without confusion. If the denominators are different, as with \( \frac{11}{4} \) and \( \frac{9}{8} \), we must first find an equivalent fraction.

• Identify the least common multiple (LCM) of the given denominators, in this case, 4 and 8.
• Convert fractions to equivalent fractions with this common denominator.

The least common denominator for 4 and 8 is 8 because 8 can be evenly divided by 4. In the exercise, \( \frac{11}{4} \) was converted to \( \frac{22}{8} \) to match the denominator of 8. With both fractions sharing a denominator, subtraction or addition becomes straightforward and logical.

Simplifying Fractions

After performing operations with fractions, it's important to ensure that the resulting fraction is in its simplest form. Simplifying a fraction means reducing it to the smallest possible numerator and denominator while keeping the same value. This makes the fraction easier to read and understand.

• Check if the numerator and denominator have any common factors.
• Divide both by the greatest common divisor (GCD) to simplify.

In some cases, like in the exercise with the result \( \frac{31}{8} \), the fraction is already in its simplest form, as 31 and 8 have no common factors other than 1. Simplifying helps in achieving a clear and efficient representation of the fraction, making it more utilitarian for further calculations or interpretations.