Problem 3 - Introductory and Intermediate Algebra for College Students 4th

Question

Convert each mixed number to an improper fraction. $$7 \frac{3}{5}$$

Step-by-Step Solution

Answer

The mixed number \(7 \frac{3}{5}\) converts to the improper fraction \( \frac{38}{5} \)

Step 1: Multiply the Whole Number by the Denominator

Firstly, multiply the whole number, which is 7, by the denominator of the fraction, which is 5. The calculation is 7*5=35.

Step 2: Add the Numerator

Then, add the result from Step 1 to the numerator of the fraction. The numerator here is 3. The calculation is 35+3=38.

Step 3: Write the Result as the Numerator Over the Denominator

Finally, write the result from Step 2 as the numerator over the denominator. The denominator is still 5. Thus, the resulting improper fraction is \( \frac{38}{5} \).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. In the example provided, 7\( \frac{3}{5} \) is a mixed number. Here, 7 represents the whole number part, while \( \frac{3}{5} \) is the fractional part.

Understanding mixed numbers is important because they show quantities that are more than a whole but less than the next whole number. They are often used in everyday situations, like measuring distances or cooking.

Some key points to remember about mixed numbers are:

• They always include a whole number and a fraction.
• The fraction in a mixed number is always a proper fraction, meaning the numerator is smaller than the denominator.
• Mixed numbers can easily be converted into improper fractions for calculations.

Recognizing mixed numbers helps in identifying values that are not whole and need further breakdown to understand their complete value.

Improper Fractions

Improper fractions are fractions where the numerator (top number) is greater than or equal to the denominator (bottom number). In simple terms, this means that the fraction is equivalent to, or more than, one whole.

In our example, the improper fraction \( \frac{38}{5} \) resulted from converting the mixed number 7\( \frac{3}{5} \). Here, the numerator 38 is greater than the denominator 5.

Why are improper fractions useful? They provide a convenient way to deal with fractional values in mathematical operations. Here are some notable aspects:

• They can be formed by multiplying the whole number with the fraction's denominator and adding the numerator.
• They are simplified forms for calculations, especially in operations like multiplication or division.
• Though they might look unusual, they are handled just like any other fractions in solution steps.

Understanding improper fractions opens the gateway to manipulating and simplifying complex mathematical problems.

Numerator and Denominator Conversion

Conversion between mixed numbers and improper fractions hinges on the manipulation of numerators and denominators.

To convert a mixed number to an improper fraction, you follow a simple three-step process:
1. **Multiply** the whole number by the denominator of the fraction to get an intermediate result.
2. **Add** this result to the numerator of the fraction to find the new numerator.
3. **Express** the fraction with the new numerator and the original denominator.

For example, to convert 7\( \frac{3}{5} \) into an improper fraction:

• Multiply 7 (whole number) by 5 (denominator) to get 35.
• Add 3 (numerator) to 35 to get 38.
• The improper fraction becomes \( \frac{38}{5} \).

This straightforward conversion helps streamline calculations and provides greater flexibility in solving mathematical problems.