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Chapter 5: Problem 31

Write the improper fraction as a mixed number. $$\frac{103}{-8}$$

Short Answer

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Question: Rewrite the improper fraction with a negative denominator, $\frac{103}{-8}$, as a mixed number. Answer: The mixed number representation of the given fraction is $-12-\frac{7}{8}$.

Step by step solution

01

Determine the whole number part

Divide the numerator by the denominator to get the quotient and remainder. This will give us the whole number and the proper fraction part. $$\frac{103}{-8} = quotient + \frac{remainder}{-8}$$
02

Calculate the quotient and remainder

To calculate the quotient and remainder, perform the division: $$103\div -8 = -12\,remainder\,7$$ So, the quotient is -12 and the remainder is 7.
03

Write the mixed number

Using the quotient and remainder from step 2, we can form the mixed number: $$-12+\frac{7}{-8}$$
04

Simplify the mixed number

We got a proper fraction with a negative denominator, we can simplify it since the whole number part is already negative: $$-12-\frac{7}{8}$$ Now, we have the mixed number representing the initial fraction.

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Key Concepts

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

Mixed Numbers
A mixed number is a combination of a whole number and a proper fraction. It is often used to express improper fractions in a more easy-to-understand manner. An improper fraction contains a numerator greater than the denominator, which sometimes makes them difficult to interpret at a glance.
To rewrite an improper fraction into a mixed number:
  • First, divide the numerator by the denominator to find the whole number portion.

  • The quotient from this division process represents the whole number part of the mixed number.

  • The remainder becomes the numerator of the fraction part, while the denominator remains the same.

Mixed numbers are helpful in making numbers easier to read and understand, especially in practical situations like cooking or measuring.
Division
Division is the mathematical process where you determine how many times a divisor can fit into another number, known as the dividend. When dealing with fractions, division helps in splitting a fraction into its components.In the case of converting an improper fraction, like \(\frac{103}{-8}\), into a mixed number:
  • Divide 103 by -8.

  • Find the quotient to serve as the whole number of the mixed number.

  • Use the remainder to form the proper fraction part.

This division process differs slightly from standard division, as unit consistency must be maintained for the entire fraction.
Quotient and Remainder
In any division calculation, the quotient is the result of the division, and the remainder is what's left over when the division is performed. For example, when you divide 103 by -8:
  • The quotient is -12, indicating how many times -8 fits into 103.

  • The remainder is 7, representing what is left over after taking out full groups of -8.

These components are crucial for reconstructing an improper fraction as a mixed number. The quotient becomes the whole part, and the remainder helps to form the fractional part of the mixed number.
Simplifying Fractions
Simplifying fractions means making the fraction as simple as possible. We do this by ensuring both numerator and denominator are in their smallest whole number form.In a mixed number like \(-12+\frac{7}{-8}\), simplifying is necessary:
  • The negative sign should be consistent across the number to properly represent it.

  • Here, \(-12+\frac{7}{-8}\) becomes \(-12-\frac{7}{8}\).

Remember, a mixed number is simple when it has a positive fraction and a whole number of the same sign, ensuring clarity in what the number actually represents.

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Most popular questions from this chapter

Solve. Write all answers in lowest terms. \(\frac{5}{6}\) of a company's employees live within a 15 -mile radius of the company. Of these employees, \(\frac{3}{4}\) live within a 10 -mile radius. What fraction of all employees live within a 10 -mile radius?

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Use \(<,>,\) or \(=\) to make a true statement. $$2 \frac{5}{6} ? 2 \frac{10}{12}$$

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Solve. Write all answers in lowest terms. A single share of a certain computer company's stock is listed at \(\$ 6 \frac{3}{8}\) per share. What is the total value of 500 shares of the stock?
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