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Magazine Chapter 5: Problem 37
Find each difference. Write in simplest form. $$-19 \frac{3}{8}-\left(-4 \frac{3}{4}\right)$$
Short Answer
Expert verified
The result is \(-\frac{117}{8}\).
Step by step solution
01
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 \(3\):\[-4 \frac{3}{4} = -\frac{(4 \times 4) + 3}{4} = -\frac{16 + 3}{4} = -\frac{19}{4}\]
02
Change Subtraction to Addition
Rewrite the expression by changing the subtraction of a negative to addition:\[-\frac{155}{8} - (-\frac{19}{4}) = -\frac{155}{8} + \frac{19}{4}\]
03
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:\[\frac{19}{4} = \frac{19 \times 2}{4 \times 2} = \frac{38}{8}\]
04
Add the Fractions
Now that the fractions have a common denominator, add them:\[-\frac{155}{8} + \frac{38}{8} = \frac{-155 + 38}{8} = \frac{-117}{8}\]
05
Simplify the Result
The fraction \(\frac{-117}{8}\) is already in simplest form, as \(117\) and \(8\) do not have a common factor.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
Converting \(\frac{19}{4}\) to \(\frac{38}{8}\) allowed these fractions to be added directly, simplifying the entire process.
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.
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.
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.
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).
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.
