Problem 81
Question
Find the reciprocal. \(-8 \frac{3}{4}\)
Step-by-Step Solution
Verified Answer
The reciprocal of \(-8 \frac{3}{4}\) is \(-\frac{4}{35}\).
1Step 1: Convert Mixed Number to Improper Fraction
Firstly, the mixed number has to be converted to an improper fraction. The process: multiply the whole number by the denominator, add the numerator, and write the resultant over the denominator. \(-8 \frac{3}{4}= -\frac{35}{4}\)
2Step 2: Find the Reciprocal
The reciprocal of a fraction is obtained by interchanging the numerator and the denominator. So, the reciprocal of \(-\frac{35}{4}\) is \(-\frac{4}{35}\).
Key Concepts
Converting Mixed Numbers to Improper FractionsImproper FractionNegative Fractions
Converting Mixed Numbers to Improper Fractions
Understanding how to convert mixed numbers to improper fractions is a fundamental skill in mathematics. A mixed number consists of a whole number and a proper fraction, whereas an improper fraction has a numerator that is larger than or equal to its denominator.
To convert a mixed number to an improper fraction, you follow these steps: First, multiply the whole number by the denominator of the fraction part. Then, add this result to the numerator of the fraction part. Lastly, place this sum over the original denominator. For instance, consider the mixed number (-8 \( \frac{3}{4} \)). To convert it, multiply 8 by 4 (the denominator), getting 32, then add 3 (the numerator), resulting in 35. Thus, the improper fraction form is (-\( \frac{35}{4} \)).
Converting a mixed number to an improper fraction allows for easier and more consistent application of arithmetic operations, such as addition, subtraction, multiplication, and finding reciprocals. It's a crucial step before proceeding to any calculations.
To convert a mixed number to an improper fraction, you follow these steps: First, multiply the whole number by the denominator of the fraction part. Then, add this result to the numerator of the fraction part. Lastly, place this sum over the original denominator. For instance, consider the mixed number (-8 \( \frac{3}{4} \)). To convert it, multiply 8 by 4 (the denominator), getting 32, then add 3 (the numerator), resulting in 35. Thus, the improper fraction form is (-\( \frac{35}{4} \)).
Converting a mixed number to an improper fraction allows for easier and more consistent application of arithmetic operations, such as addition, subtraction, multiplication, and finding reciprocals. It's a crucial step before proceeding to any calculations.
Improper Fraction
An improper fraction occurs when the numerator is greater than or equal to the denominator. Unlike proper fractions, where the numerator is less than the denominator, improper fractions represent quantities greater than or equal to one. The improper fraction showcases how many times the denominator fits into the numerator.
For example, the fraction \( \frac{35}{4} \) signifies that four goes into thirty-five eight times with a remainder. In more visual terms, if you have 35 pieces of cake and you group them into sets of 4, you would have 8 sets with 3 pieces left over – hence, the mixed number -8 \( \frac{3}{4} \) or the improper fraction -\( \frac{35}{4} \). Understanding this concept is invaluable when performing operations involving fractions and is especially necessary when you begin working with algebraic expressions that include fractions.
For example, the fraction \( \frac{35}{4} \) signifies that four goes into thirty-five eight times with a remainder. In more visual terms, if you have 35 pieces of cake and you group them into sets of 4, you would have 8 sets with 3 pieces left over – hence, the mixed number -8 \( \frac{3}{4} \) or the improper fraction -\( \frac{35}{4} \). Understanding this concept is invaluable when performing operations involving fractions and is especially necessary when you begin working with algebraic expressions that include fractions.
Negative Fractions
Fractions can also be negative, which can occur in any mathematical context where division is applied to negative numbers. A negative fraction means that the value represented is less than zero. It's important to recognize that the negative sign can be associated with the numerator, the denominator, or the entire fraction, yet mathematically, they result in the same value.
For example, the negative mixed number -8 \( \frac{3}{4} \) can be converted to an improper fraction as -\( \frac{35}{4} \). This negative sign indicates that the value of the fraction is less than zero. A common mistake students make is to misplace the negative sign when converting between mixed numbers, improper fractions, or reciprocals, which can lead to incorrect answers. The location of the negative sign doesn't alter the magnitude of the fraction, just its direction on the number line.
For example, the negative mixed number -8 \( \frac{3}{4} \) can be converted to an improper fraction as -\( \frac{35}{4} \). This negative sign indicates that the value of the fraction is less than zero. A common mistake students make is to misplace the negative sign when converting between mixed numbers, improper fractions, or reciprocals, which can lead to incorrect answers. The location of the negative sign doesn't alter the magnitude of the fraction, just its direction on the number line.
Other exercises in this chapter
Problem 81
Simplify the expression. $$ \sqrt{10} \cdot \sqrt{20} $$
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Add. Write the answer as a mixed number in simplest form. $$ 2 \frac{1}{2}+\frac{5}{7} $$
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Tell whether the equation has two solutions, one solution, or no real solution. $$ 4 x^{2}+4 x+1=0 $$
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List all the factors of the number. 36
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