Problem 68
Question
Divide. See Example 5. $$ -\frac{2}{3} \div \frac{7}{8} $$
Step-by-Step Solution
Verified Answer
The result is \(-\frac{16}{21}\).
1Step 1: Write the Division Problem as a Multiplication
The first step in dividing fractions is to rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor. The given problem is \(-\frac{2}{3} \div \frac{7}{8}\). To do this, multiply by the reciprocal of \(\frac{7}{8}\), which is \(\frac{8}{7}\). Thus, we write:\[-\frac{2}{3} \times \frac{8}{7}.\]
2Step 2: Multiply the Numerators
Now that we have a multiplication problem, multiply the numerators of the two fractions. The numerators are \(-2\) and \(8\). So, the product of the numerators is:\[-2 \times 8 = -16.\]
3Step 3: Multiply the Denominators
Next, multiply the denominators of the two fractions. The denominators are \(3\) and \(7\). So, the product of the denominators is:\[3 \times 7 = 21.\]
4Step 4: Write the Resulting Fraction
Combine the results of the numerator and the denominator multiplication to form a new fraction:\[\frac{-16}{21}.\]
5Step 5: Simplify the Fraction (if necessary)
Finally, check if the fraction \(\frac{-16}{21}\) can be simplified. Since 16 and 21 have no common factors other than 1, the fraction is already in its simplest form.
Key Concepts
ReciprocalMultiplying FractionsNumerators and Denominators
Reciprocal
Understanding the concept of a reciprocal is crucial when dividing fractions. The reciprocal of a fraction is simply what you get when you flip the fraction upside down. For example, if you have the fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). This is an important step in the process because when you divide by a fraction, it's the same as multiplying by its reciprocal.
To clarify this, look at the division problem \(-\frac{2}{3} \div \frac{7}{8}\). Instead of dividing directly, we convert this to a multiplication problem by taking the reciprocal of \(\frac{7}{8}\), which is \(\frac{8}{7}\). This allows us to multiply \(-\frac{2}{3} \times \frac{8}{7}\) instead. Remember:
To clarify this, look at the division problem \(-\frac{2}{3} \div \frac{7}{8}\). Instead of dividing directly, we convert this to a multiplication problem by taking the reciprocal of \(\frac{7}{8}\), which is \(\frac{8}{7}\). This allows us to multiply \(-\frac{2}{3} \times \frac{8}{7}\) instead. Remember:
- The reciprocal flips the numerator and denominator.
- Helps to convert division into multiplication.
- Simplifies the division process of fractions.
Multiplying Fractions
Now that you have transformed the division problem into a multiplication one using the reciprocal, you can proceed with multiplying fractions. Multiplying fractions is straightforward and involves two main steps: multiplying the numerators and multiplying the denominators.
Here's what you do step-by-step:
Here's what you do step-by-step:
- Multiply the top numbers (numerators) together. In our example, you multiply \(-2\) by \(8\), which equals \(-16\).
- Next, multiply the bottom numbers (denominators) together. For this, you multiply \(3\) by \(7\), resulting in \(21\).
- Combine these two results to form a new fraction: \(\frac{-16}{21}\).
- Multiply numerator with numerator.
- Multiply denominator with denominator.
- Combine results to form a new fraction.
Numerators and Denominators
To fully grasp how to divide fractions, it's essential to understand numerators and denominators. Fractions consist of two parts: the numerator, which is the top number, and the denominator, the bottom number. They play Vital roles in fraction operations.
Numerator: Represents how many parts of the whole or set you have. In \(\frac{2}{3}\), the \(2\) is the numerator, indicating 2 parts out of 3.
Denominator: Indicates into how many equal parts the whole is divided. In our example, \(3\) is the denominator, showing that we divide the whole into 3 equal parts.
When dividing fractions, multiply these correctly as discussed in the multiplication section:
Numerator: Represents how many parts of the whole or set you have. In \(\frac{2}{3}\), the \(2\) is the numerator, indicating 2 parts out of 3.
Denominator: Indicates into how many equal parts the whole is divided. In our example, \(3\) is the denominator, showing that we divide the whole into 3 equal parts.
When dividing fractions, multiply these correctly as discussed in the multiplication section:
- To multiply, combine numerators and denominators from each fraction separately.
- Understanding these parts helps simplify and solve fraction problems efficiently.
- Remember, maintaining the order of multiplication contributes to an accurate result.