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Chapter 7: Problem 97

Add or subtract as indicated. $$ -\frac{3}{4}+\frac{1}{12} $$

Short Answer

Expert verified
\( -\frac{2}{3} \).

Step by step solution

01

Find a common denominator

The denominators are 4 and 12. The least common denominator (LCD) is 12 since 12 is the smallest number that both 4 and 12 can divide into without leaving a remainder.
02

Convert fractions

Convert \(\frac{-3}{4}\) to a fraction with a denominator of 12. Multiply both the numerator and the denominator by 3: \(\frac{-3 \times 3}{4 \times 3} = \frac{-9}{12}\).
03

Write the equation with the common denominator

Now, write the equation \(-\frac{3}{4} + \frac{1}{12}\) using 12 as the common denominator: \(\frac{-9}{12} + \frac{1}{12}\).
04

Add or subtract the numerators

Since the denominators are the same, add the numerators: \(-9 + 1 = -8\). This gives \(\frac{-8}{12}\).
05

Simplify the fraction

Simplify \(\frac{-8}{12}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4: \(\frac{-8 \/ 4}{12 \/ 4} = \frac{-2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

least common denominator
When adding or subtracting fractions, the first step is to find a common denominator. The **least common denominator** (LCD) is the smallest number that can be evenly divided by both denominators. In our example, we have fractions with denominators of 4 and 12.
Finding the least common denominator:
  • List the multiples of each denominator.
  • The multiples of 4 are: 4, 8, 12, 16, 20, ...
  • The multiples of 12 are: 12, 24, 36, 48, ...
  • The smallest common multiple of 4 and 12 is 12.
Using the least common denominator ensures we can combine the fractions easily.
fraction simplification
After performing operations with fractions, it's crucial to simplify the result. **Fraction simplification** means reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, if you have \(\frac{-8}{12}\):
  • Find the GCD of 8 and 12, which is 4.
  • Divide the numerator and the denominator by the GCD.
  • \(\frac{-8 \/ 4}{12 \/ 4} = \frac{-2}{3}\)
The simplified fraction \(\frac{-2}{3}\) is easier to understand and work with.
numerator and denominator
In any fraction, the **numerator** and **denominator** are its essential elements. The numerator is the top part of the fraction, showing how many parts we have. The denominator is the bottom part, showing how many parts make up a whole.
Consider \(\frac{-3}{4}\):
  • The numerator is -3, indicating we have -3 parts.
  • The denominator is 4, meaning a whole consists of 4 parts.
By understanding what numerators and denominators represent, you can better interpret and manipulate fractions in operations like addition and subtraction.

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Most popular questions from this chapter

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5}{x+2}+\frac{2}{x^{2}-2 x+4}-\frac{60}{x^{3}+8} $$

Use personal experience or intuition to determine whether the situation suggests direct or inverse variation. Your age and the probability that you believe in Santa Claus

As explained in this section, any values that would cause a denominator to equal 0 must be exchaded from the domain and, consequently, as solutions of an equation that has variable expressions in the denominators. (a) Without actually solving each equation, list all possible values that would have to be rejected if they appeared as proposed solutions. (b) Then give the domain, using set-builder notation. $$ \frac{6}{4 x+7}-\frac{3}{x}=\frac{5}{6 x-13} $$

Multiply or divide as indicated. $$ \frac{5 a^{4} b^{2}}{16 a^{2} b} \div \frac{25 a^{2} b}{60 a^{3} b^{2}} $$

Multiply or divide as indicated. $$ \left(\frac{2 x^{3}+3 x^{2}-2 x}{3 x-15} \div \frac{2 x^{3}-x^{2}}{x^{2}-3 x-10}\right) \cdot \frac{5 x^{2}-10 x}{3 x^{2}+12 x+12} $$
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