Problem 89 - Elementary Algebra

Question

For exercises 81-96, evaluate. $$ -\frac{1}{4}-\frac{5}{6} $$

Step-by-Step Solution

Answer

-\frac{13}{12}

Step 1 - Find a Common Denominator

To subtract \( \- \frac{1}{4} - \frac{5}{6} \), first find a common denominator for the fractions. The denominators are 4 and 6. The least common denominator (LCD) is 12 since it is the smallest number divisible by both 4 and 6.

Step 2 - Convert to Equivalent Fractions

Convert each fraction to an equivalent fraction with the common denominator 12. \( \- \frac{1}{4} = \- \frac{1 \cdot 3}{4 \cdot 3} = \- \frac{3}{12} \) and \( \frac{5}{6} = \frac{5 \cdot 2}{6 \cdot 2} = \frac{10}{12} \).

Step 3 - Subtract the Fractions

Subtract the numerators of the two fractions, keeping the common denominator: \( \- \frac{3}{12} - \frac{10}{12} = \frac{-3 - 10}{12} = \frac{-13}{12} \).

Key Concepts

common denominator

When subtracting fractions, it's essential to have a common denominator. This is because fractions must be expressed with the same denominator to be combined. In the given exercise, the denominators are 4 and 6. The Least Common Denominator (LCD) is the smallest number that both denominators can divide into without leaving a remainder. To find the LCD, list the multiples of each denominator:

• Multiples of 4: 4, 8, 12, 16, 20, 24,...
• Multiples of 6: 6, 12, 18, 24, 30, 36,...

The smallest common multiple is 12. Thus, we use 12 as our common denominator. Finding the common denominator allows us to convert fractions so they can be easily subtracted.

equivalent fractions

An equivalent fraction is a fraction that represents the same value when both the numerator and denominator are multiplied or divided by the same non-zero number. To convert fractions to equivalent fractions with a common denominator in the given exercise, we do the following:

• Convert \(\-\frac{1}{4}\) to a fraction with 12 as the denominator. Since 12 is 4 multiplied by 3, multiply both the numerator and denominator by 3: \(\-\frac{1\cdot3}{4\cdot3}=\-\frac{3}{12}\).
• Convert \(\frac{5}{6}\) to a fraction with 12 as the denominator. Since 12 is 6 multiplied by 2, multiply both the numerator and denominator by 2: \(\frac{5\cdot2}{6\cdot2}=\frac{10}{12}\).

This conversion is critical because it allows us to subtract fractions that have different denominators by making them comparable.

subtracting fractions

Once fractions are expressed with a common denominator, subtracting them is straightforward. Simply subtract the numerators and keep the common denominator the same:

• In the given problem, we have equivalent fractions \(\-\frac{3}{12}\) and \(\frac{10}{12}\).
• Subtract the numerators: \(\-3-10=\-13\).
• Keep the common denominator of 12: \(\frac{-13}{12}\).

So, the result of \(\-\frac{1}{4}-\frac{5}{6}\) is \(\frac{-13}{12}\). Remember that the numerator indicates the sum or difference of parts, while the denominator indicates the number of equal parts the whole divides into. By understanding these steps, subtracting fractions becomes more manageable.