Problem 53 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The sum is \(-5 \frac{7}{8}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert \(-3 \frac{3}{4}\) and \(-2 \frac{1}{8}\) into improper fractions. \(-3 \frac{3}{4}\) becomes \(-\frac{15}{4}\) and \(-2 \frac{1}{8}\) becomes \(-\frac{17}{8}\).

Step 2: Find a Common Denominator

The denominators are 4 and 8. The least common denominator is 8. Convert \(-\frac{15}{4}\) to \(-\frac{30}{8}\).

Step 3: Add the Fractions

Add \(-\frac{30}{8}\) and \(-\frac{17}{8}\) to get \(-\frac{47}{8}\).

Step 4: Convert to Mixed Number

Divide the numerator by the denominator to convert \(-\frac{47}{8}\) into a mixed number: \(-5 \frac{7}{8}\).

Key Concepts

Mixed Numbers

A mixed number is a type of fraction that shows a whole number and a proper fraction together. For example, in a mixed number like \( -3 \frac{3}{4} \), the \( -3 \) is the whole number, while the \( \frac{3}{4} \) is the proper fraction. Mixed numbers make it easier to conceptualize quantities larger than one and are often more intuitive than improper fractions.

Mixed numbers are particularly useful in everyday contexts like cooking or measurement, where you might see something like 2 ½ cups or 5 ¾ inches. However, when performing arithmetic operations like addition or subtraction, it's generally easier to convert mixed numbers into improper fractions. This simplifies calculations as you'll be dealing with only fractions, not a mix of whole numbers and fractions.

• A mixed number consists of a whole number and a proper fraction.
• They are converted to improper fractions for easier arithmetic operations.
• Best used for clearer understanding in practical applications.

Improper Fractions

An improper fraction is a type of fraction where the numerator is equal to or larger than the denominator, meaning it represents a value greater than or equal to one. For example, \( \frac{15}{4} \) is an improper fraction. Converting mixed numbers to improper fractions involves multiplying the whole number by the fraction's denominator and adding the result to the numerator.

In the exercise, \( -3 \frac{3}{4} \) is converted to \( -\frac{15}{4} \) by the following: Multiply 3 (the whole number) by 4 (the denominator) and add 3 (the numerator) resulting in 15. Similarly, for \( -2 \frac{1}{8} \), multiply 2 by 8 and add 1 to get 17, leading to \( -\frac{17}{8} \). The improper fractions make it easier to find common denominators and carry out addition or subtraction.

• An improper fraction has a numerator equal to or larger than the denominator.
• Useful for performing arithmetic calculations.
• Converted from mixed numbers by multiplying the whole number by the denominator and adding the numerator.

Common Denominators

When adding or subtracting fractions, both fractions need to have the same denominator. This is referred to as having a common denominator. This ensures that the fractions are comparable and can be combined or reduced correctly. Finding a common denominator involves identifying the lowest number that both denominators can divide into evenly.

In the given exercise, the denominators 4 and 8 require conversion to a common denominator to proceed with the calculation. The least common denominator of these fractions is 8, as both 4 and 8 divide evenly into it. The fraction \( -\frac{15}{4} \) is converted to a denominator of 8 by multiplying the numerator and the denominator by 2, resulting in \( -\frac{30}{8} \). With this common denominator, the fractions can be easily added or subtracted.

• Fractions must have a common denominator for addition or subtraction.
• The least common denominator is the smallest divisible number for both denominators.
• Convert fractions by adjusting numerators and denominators to match this common value.

Simplifying Fractions

After solving a fraction problem, you may end up with a fraction that can be reduced or simplified to a more concise form. Simplifying fractions makes them easier to understand and use in further calculations. Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In the exercise, \( -\frac{47}{8} \) is already in its simplest form because there are no common factors other than 1 for 47 and 8. When simplifying or converting an improper fraction back to a mixed number, check for the greatest common divisor or any necessary adjustments that ensure the fraction is as simple as possible.

For converting an improper fraction to a mixed number, divide the numerator by the denominator to find the whole number part and use the remainder as the new numerator. Here, dividing 47 by 8 gives 5, with a remainder of 7, yielding the mixed number \( -5 \frac{7}{8} \).

• Simplifying involves dividing by the greatest common divisor of the numerator and denominator.
• Simplified fractions are easier to interpret and use.
• Convert improper fractions to mixed numbers by division: whole number with the remainder as the proper fraction.