Problem 71
Question
In Exercises \(47-76,\) perform the indicated division or state that the expression is undefined. \(-\frac{14}{9} \div \frac{7}{8}\)
Step-by-Step Solution
Verified Answer
The simplified form of the given division operation is \(-\frac{112}{63}\).
1Step 1: Identify the Reciprocal of the Divisor
The divisor in this case is \(\frac{7}{8}\). The reciprocal of a fraction is obtained by interchanging the numerator and the denominator. Therefore, the reciprocal of \(\frac{7}{8}\) will be \(\frac{8}{7}\).
2Step 2: Convert Division to Multiplication
Instead of dividing by a fraction, it could be multiplied by the reciprocal of that fraction. So, the division problem \(-\frac{14}{9} \div \frac{7}{8}\) changes to \(-\frac{14}{9} \times \frac{8}{7}\).
3Step 3: Multiply the Fractions
The multiplication of two fractions is done by multiplying the numerators together and denominators together. The multiplication should give \(-\frac{14 \times 8}{9 \times 7} = -\frac{112}{63}\).
Key Concepts
Reciprocal of a FractionConvert Division to MultiplicationMultiplication of Fractions
Reciprocal of a Fraction
Understanding the concept of a reciprocal is crucial when it comes to dividing fractions in algebra. The reciprocal of a fraction simply means flipping the fraction upside down. In other words, you switch the numerator (the top number) and the denominator (the bottom number). For instance, if we consider the fraction \( \frac{2}{3} \), its reciprocal would be \( \frac{3}{2} \).
Evaluating the reciprocal is often the first step in dividing fractions because it allows us to convert the division operation into multiplication, which is a simpler operation to perform. When we talk about a fraction like \( \frac{7}{8} \) from our original exercise, its reciprocal is \( \frac{8}{7} \). It's essential to note that every number has a reciprocal except for zero because division by zero is undefined. This is why we also need to make sure the denominator of our original fraction is not zero before we find its reciprocal.
Evaluating the reciprocal is often the first step in dividing fractions because it allows us to convert the division operation into multiplication, which is a simpler operation to perform. When we talk about a fraction like \( \frac{7}{8} \) from our original exercise, its reciprocal is \( \frac{8}{7} \). It's essential to note that every number has a reciprocal except for zero because division by zero is undefined. This is why we also need to make sure the denominator of our original fraction is not zero before we find its reciprocal.
Convert Division to Multiplication
When dividing fractions, such as \( -\frac{14}{9} \div \frac{7}{8} \), we have a technique to make the process easier—converting the division problem into a multiplication problem. This is a two-step process.
Firstly, find the reciprocal of the divisor (the second fraction). As we learned in the previous section, the reciprocal of \( \frac{7}{8} \) is \( \frac{8}{7} \). Secondly, change the division sign to a multiplication sign and multiply the dividend (the first fraction) by the reciprocal of the divisor. This changes our original division problem to \( -\frac{14}{9} \times \frac{8}{7} \). This method simplifies the computation because multiplication of fractions is more straightforward than division.
Firstly, find the reciprocal of the divisor (the second fraction). As we learned in the previous section, the reciprocal of \( \frac{7}{8} \) is \( \frac{8}{7} \). Secondly, change the division sign to a multiplication sign and multiply the dividend (the first fraction) by the reciprocal of the divisor. This changes our original division problem to \( -\frac{14}{9} \times \frac{8}{7} \). This method simplifies the computation because multiplication of fractions is more straightforward than division.
Multiplication of Fractions
Multiplying fractions may seem daunting, but it is simpler than it looks. The multiplication of fractions involves two main steps. First, multiply the numerators (the top numbers) of the fractions together. Second, multiply the denominators (the bottom numbers) together.
In our problem, \( -\frac{14}{9} \times \frac{8}{7} \), we multiply the numerators: 14 and 8, to get 112. We then multiply the denominators: 9 and 7, to get 63. Combining these, we get the product \( -\frac{112}{63} \), which is the solution to the multiplication problem.
If possible, the final step in multiplying fractions is to simplify. In this case, the fraction \( -\frac{112}{63} \) simplifies to \( -\frac{16}{9} \), since both numbers are divisible by 7. It's always good to present your answer in its simplest form to make it easier to understand.
In our problem, \( -\frac{14}{9} \times \frac{8}{7} \), we multiply the numerators: 14 and 8, to get 112. We then multiply the denominators: 9 and 7, to get 63. Combining these, we get the product \( -\frac{112}{63} \), which is the solution to the multiplication problem.
If possible, the final step in multiplying fractions is to simplify. In this case, the fraction \( -\frac{112}{63} \) simplifies to \( -\frac{16}{9} \), since both numbers are divisible by 7. It's always good to present your answer in its simplest form to make it easier to understand.
Other exercises in this chapter
Problem 70
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Use the order of operations to simplify each expression. $$\frac{\frac{7}{9}-3}{\frac{5}{6}} \div \frac{3}{2}+\frac{3}{4}$$
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Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. six times the product of 4 and a number
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