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Chapter 4: Problem 34

In the following exercises, rewrite the improper fraction as a mixed number. $$\frac{42}{13}$$

Short Answer

Expert verified
The mixed number is 3 \(\frac{3}{13}\).

Step by step solution

01

Divide the numerator by the denominator

Divide the numerator (42) by the denominator (13) to find the quotient and the remainder. Perform the division: \(42 \div 13 = 3 \) with a remainder of \(42 - (3 \times 13) = 42 - 39 = 3\)
02

Write the quotient as the whole number

The quotient from the division operation is 3, which will be the whole number part of the mixed number.
03

Write the remainder as the numerator

The remainder from the division operation is 3, which will be the numerator of the fractional part of the mixed number. The denominator remains the same as in the original fraction.
04

Combine the whole number and fractional part

Combine the whole number part and the fractional part to form the mixed number: \(3 \frac{3}{13}\)

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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 fraction. They are useful for expressing quantities that are between whole numbers. For example, if you have \(\frac{42}{13}\), dividing 42 by 13 gives you 3 whole parts and a remainder. Therefore, the improper fraction \(\frac{42}{13}\) can be expressed as a mixed number: \(3 \frac{3}{13}\). This representation offers a clearer picture of the quantity being described.

To rewrite an improper fraction as a mixed number:
  • Perform the division to find the whole number part.
  • The remainder becomes the numerator of the fractional part.
  • The denominator stays the same.

In our example, \(42 \div 13 = 3\) with a remainder of 3. So, \(3 \frac{3}{13}\) is the mixed number.
Division Operations
Division operations are fundamental in converting improper fractions to mixed numbers. When you divide the numerator by the denominator, you obtain two key pieces of information: the quotient and the remainder. The quotient will become the whole number of the mixed number, and the remainder will form the numerator of the fractional part.

In the case of \(\frac{42}{13}\):
  • Divide 42 by 13
  • The quotient is 3 (the whole number part).
  • The remainder is 3 (the numerator of the fraction).
  • The denominator (13) remains the same.

This results in \(3 \frac{3}{13}\).

Understanding division operations helps in simplifying complex fractions and accurately expressing quantities.
Fractions
Fractions represent parts of a whole. They consist of a numerator (the top part) and a denominator (the bottom part). An improper fraction has a numerator larger than its denominator, indicating a value greater than one. Converting these into mixed numbers makes them easier to work with and understand.

For example, \(\frac{42}{13}\) is an improper fraction. By performing the division:
  • 42 divided by 13 equals 3 (the whole number part).
  • The remainder is 3, which will be the numerator in the fractional part.
  • The denominator remains as 13.
This converts \(\frac{42}{13}\) into \(3 \frac{3}{13}\).

Proper handling of fractions is crucial for solving mathematical problems efficiently. Understanding how to convert between different forms helps in various applications.

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Most popular questions from this chapter

Evaluate. $$ x+\frac{1}{3} $$ $$ (a)x=\frac{2}{3} $$ $$ (b)x=-\frac{5}{6} $$

In the following exercises, perform the indicated operation. $$ 9 \frac{13}{20}-4 \frac{11}{20} $$

In the following exercises, change to equivalent fractions using the given LCD. $$-\frac{9}{16} \text { and } \frac{5}{12}, \mathrm{LCD}=48$$

In the following exercises, perform the indicated operation. $$ 2 \frac{11}{12}-1 \frac{7}{12} $$

In the following exercises, solve the equation. Translate and solve: The quotient of \(p\) and \(-4\) is \(-8 .\) Solve for \(p\)
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