Problem 15 Prep Exercise 2 Explain how to d... [FREE SOLUTION]

Chapter 5: Problem 15

Prep Exercise 2 Explain how to divide fractions. For Exercises $9-20,$ divide. Write the quotient in lowest terms. $$\frac{-7}{12} \div(-14)$$

Short Answer

Expert verified

Answer: The quotient is $-\frac{1}{24}$.

Step by step solution

Write the divisor as a fraction

We need to write the divisor, $(-14)$, as a fraction. To do this, we put $(-14)$ as the numerator and $1$ as the denominator, giving us: $$(-14) = \frac{-14}{1}$$

Finding the reciprocal of the divisor

Next, we need to find the reciprocal of the divisor, $(-14/1)$. The reciprocal of a fraction is obtained by switching the numerator and the denominator. Therefore, the reciprocal of $(-14/1)$ is: $$\frac{1}{-14}$$

Multiplying the dividend by the reciprocal of the divisor

Now that we have the reciprocal of the divisor, we can perform the division by multiplying the dividend, $(-7/12)$, by the reciprocal of the divisor, $(1/-14)$. This gives us: $$\frac{-7}{12} \div(-14) = \frac{-7}{12} \cdot \frac{1}{-14}$$

Simplifying the multiplied fractions

To simplify the multiplied fractions, we look for common factors between the numerators and denominators. We can see that $7$ is a common factor of the numerator $(-7)$ and the denominator $(-14)$. We can simplify by dividing both by $7$: $$\frac{-7}{12} \cdot \frac{1}{-14} = \frac{-1\cdot7}{12} \cdot \frac{1}{-2\cdot7}$$ Now, we can cancel out the common factor $7$: $$\frac{-1}{12} \cdot \frac{1}{-2}$$

Multiplying the simplified fractions

Now, we multiply the simplified fractions by multiplying the numerators together and the denominators together: $$\frac{-1}{12} \cdot \frac{1}{-2} = \frac{(-1)(1)}{(12)(-2)}$$

Calculating the final quotient

Finally, we perform the multiplication to find the quotient: $$\frac{(-1)(1)}{(12)(-2)} = \frac{1}{-24}$$ However, since a negative in the denominator is not a standard form, we can rewrite it as: $$\frac{1}{-24} = -\frac{1}{24}$$

Answer

The quotient of the given division problem, $\frac{-7}{12} \div(-14)$, is: $$-\frac{1}{24}$$

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Key Concepts

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

Reciprocal

Understanding how to divide fractions starts with the idea of a *reciprocal*. The reciprocal of a number or fraction is what you multiply it by to get 1. For a fraction like $ \frac{a}{b} $, its reciprocal is $ \frac{b}{a} $. Here's why reciprocals are essential in fraction division:

Instead of dividing by a fraction, you multiply by its reciprocal. This turns a division problem into a multiplication problem, which can be easier to solve.
Finding the reciprocal involves simply flipping the numerator and the denominator of the fraction.

For example, if you have to divide $ \frac{-7}{12} $ by $ -14 $, first write $ -14 $ as $ \frac{-14}{1} $. Then, find the reciprocal, which is $ \frac{1}{-14} $. By doing this, you transform the division into a multiplication: $ \frac{-7}{12} \times \frac{1}{-14} $. This process simplifies the problem significantly.

Simplifying Fractions

After finding the reciprocal and setting up the multiplication, the next step is *simplifying fractions*. When fractions have numbers that can be divided by the same factor, they can be simplified.The process involves:

Identifying common factors in the numerator and the denominator.
Dividing both the numerator and the denominator by these common factors to reduce the fraction to its simplest form.

In our example, multiplying $ \frac{-7}{12} $ by $ \frac{1}{-14} $, note that both $-7$ and $-14$ have $7$ as a common factor. By dividing both by $7$, we simplify to $ \frac{-1}{12} \times \frac{1}{-2} $. Simplifying helps in keeping calculations neat and manageable.

Common Factors

*Common factors* are numbers that can divide two or more numbers without leaving a remainder. Recognizing these in fractions is key to simplification.Here's how you can apply this:

Look at both the numerator and the denominator to determine the largest number that divides both without a remainder. This number is the greatest common factor (GCF).
Use this GCF to simplify the fraction by dividing both the numerator and denominator by this number.
This results in a fraction that's in its simplest, most reduced form.

In the division example, the numerators and denominators share common factors that were simplified before final multiplication. This step ensures that the final result, $ -\frac{1}{24} $, comes out clean and simple, free of larger numbers that could complicate the final answer.

91影视

Prep Exercise 2 Explain how to divide fractions. For Exercises \(9-20,\) divide. Write the quotient in lowest terms. $$\frac{-7}{12} \div(-14)$$

Short Answer

Step by step solution

Write the divisor as a fraction

Finding the reciprocal of the divisor

Multiplying the dividend by the reciprocal of the divisor

Simplifying the multiplied fractions

Multiplying the simplified fractions

Calculating the final quotient

Answer

Key Concepts

Reciprocal

Simplifying Fractions

Common Factors

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