Problem 37 - Prealgebra

Question

Find each difference. Write in simplest form. $$-19 \frac{3}{8}-\left(-4 \frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The result is \(-\frac{117}{8}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers to improper fractions. For \(-19 \frac{3}{8}\), multiply \(19\) by \(8\) and add \(3\):\[-19 \frac{3}{8} = -\frac{(19 \times 8) + 3}{8} = -\frac{152 + 3}{8} = -\frac{155}{8}\]For \(-4 \frac{3}{4}\), multiply \(4\) by \(4\) and add \(3\):\[-4 \frac{3}{4} = -\frac{(4 \times 4) + 3}{4} = -\frac{16 + 3}{4} = -\frac{19}{4}\]

Step 2: Change Subtraction to Addition

Rewrite the expression by changing the subtraction of a negative to addition:\[-\frac{155}{8} - (-\frac{19}{4}) = -\frac{155}{8} + \frac{19}{4}\]

Step 3: Find a Common Denominator

Find the least common denominator (LCD) of \(8\) and \(4\), which is \(8\). Convert \(\frac{19}{4}\) to have this denominator:\[\frac{19}{4} = \frac{19 \times 2}{4 \times 2} = \frac{38}{8}\]

Step 4: Add the Fractions

Now that the fractions have a common denominator, add them:\[-\frac{155}{8} + \frac{38}{8} = \frac{-155 + 38}{8} = \frac{-117}{8}\]

Step 5: Simplify the Result

The fraction \(\frac{-117}{8}\) is already in simplest form, as \(117\) and \(8\) do not have a common factor.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of an integer and a proper fraction. For example, in the mixed number \(-19 \frac{3}{8}\), \(-19\) is the integer part and \(\frac{3}{8}\) is the fractional part.
Mixed numbers provide an easier way to express numbers greater than 1 when dealing with fractions in everyday life.

• They are often used in measurements and in situations where whole units are involved alongside fractions.

To work with mixed numbers in mathematical operations, such as addition or subtraction, it's necessary to convert them into improper fractions. This makes calculations more manageable and ensures that fractions are combined effectively.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, \(\frac{155}{8}\) is an improper fraction.

• They represent amounts greater than or equal to one whole.
• Improper fractions are beneficial for mathematical operations because they allow for straightforward arithmetic processes.

To convert a mixed number to an improper fraction, multiply the whole number by the fraction's denominator, add the numerator, and place this sum over the original denominator. For instance, \(-19 \frac{3}{8}\) becomes \(-\frac{155}{8}\). This process streamlines calculation and provides a uniform framework for working with fractions.

Common Denominator

Finding a common denominator is a crucial step when dealing with multiple fractions, especially in addition or subtraction.
This commonality helps compare fractions directly by making their denominators the same.

• The least common denominator (LCD) is the smallest number that each of the denominators can divide into evenly.
• It minimizes the calculations necessary to find a common basis for comparison.

For instance, fraction \(\frac{19}{4}\) needed to match the denominator of \(-\frac{155}{8}\), which required finding a common denominator of 8.
Converting \(\frac{19}{4}\) to \(\frac{38}{8}\) allowed these fractions to be added directly, simplifying the entire process.

Fraction Simplification

Fraction simplification means reducing a fraction to its simplest form where the numerator and denominator have no common factors other than 1.
A fraction is in simplest form when it cannot be further reduced.

• This process makes fractions easier to work with and understand.
• Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In this exercise, the solution \(\frac{-117}{8}\) was already in simplest form.
The numbers 117 and 8 do not share any factors other than 1, confirming that no further reduction was possible. By ensuring fractions are simplified, comparisons and calculations become clearer and more efficient.