Question

\- A mixed number represents the sum of an integer and a proper fraction. This sum is implied, meaning that the sum sign, "+," is not written between the integer and the fraction. For example, the positive mixed number $4 \frac{3}{7}\( represents \)4+\frac{3}{7}\( and the negative mixed number \)-4 \frac{3}{7}$ represents the sum of a negative integer and a negative proper fraction, \(-4+\left(-\frac{3}{7}\right)\) or \(-4-\frac{3}{7}\).

Short answer

Question: Provide an example of a mixed number and convert it to an improper fraction and then back to a mixed number. Include a negative example as well. Answer: Example 1 (Positive mixed number): Mixed number \(5\frac{2}{3}\). Convert to improper fraction: \(\frac{17}{3}\). Convert back to mixed number: \(5\frac{2}{3}\). Example 2 (Negative mixed number): Mixed number \(-2\frac{5}{6}\). Convert to improper fraction: \(\frac{-17}{6}\) or \(-\frac{17}{6}\). Convert back to mixed number: \(-2\frac{5}{6}\).

Step-by-step solution

Understanding Mixed Numbers and Improper Fractions

A mixed number is the sum of an integer and a proper fraction, while an improper fraction is a fraction where the numerator is equal to or greater than the denominator. To convert between the two, we can use the relationship between mixed numbers and improper fractions.

Converting a Mixed Number to an Improper Fraction

To convert a mixed number to an improper fraction, follow these steps: 1. Multiply the whole number by the denominator of the fraction. 2. Add the numerator to the result from step 1. 3. Write the result from step 2 as the new numerator over the original denominator. For example, convert \(4\frac{3}{7}\) to an improper fraction: 1. \(4 \times 7 = 28\) 2. \(28 + 3 = 31\) 3. The improper fraction is \(\frac{31}{7}\).

Converting an Improper Fraction to a Mixed Number

To convert an improper fraction to a mixed number, follow these steps: 1. Divide the numerator by the denominator. 2. Write the quotient as the whole number part of the mixed number. 3. Write the remainder as the numerator and the original denominator as the denominator of the proper fraction part of the mixed number. For example, convert \(\frac{31}{7}\) to a mixed number: 1. \(31 \div 7 = 4\) with a remainder of \(3\) 2. The whole number is \(4\). 3. The proper fraction part is \(\frac{3}{7}\). 4. The mixed number is \(4\frac{3}{7}\).

Converting Negative Mixed Numbers to Improper Fractions

To convert a negative mixed number to an improper fraction, use the same steps as in Step 2, but ensure the negative sign is carried through the calculation. For example, convert \(-4\frac{3}{7}\) to an improper fraction: 1. \(-4 \times 7 = -28\) 2. \(-28 - 3 = -31\) 3. The improper fraction is \(\frac{-31}{7}\) or \(-\frac{31}{7}\).

Converting Negative Improper Fractions to Mixed Numbers

To convert a negative improper fraction to a mixed number, use the same steps as in Step 3, but ensure the negative sign is included in the whole number part. For example, convert \(\frac{-31}{7}\) or \(-\frac{31}{7}\) to a mixed number: 1. \(-31 \div 7 = -4\) with a remainder of \(3\) 2. The whole number is \(-4\). 3. The proper fraction part is \(\frac{3}{7}\). 4. The mixed number is \(-4\frac{3}{7}\).