Problem 76 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 7 \cdot 1 \frac{3}{28} $$

Step-by-Step Solution

Answer

\( 7 \cdot 1 \frac{3}{28} = \frac{217}{28} \)

Step 1: Convert Mixed Number to Improper Fraction

The first step is to convert the mixed number into an improper fraction for easier multiplication. We have:\[ 1 \frac{3}{28} = \frac{28}{28} + \frac{3}{28} = \frac{31}{28} \]

Step 2: Multiply Whole Number by Improper Fraction

Now, multiply the whole number 7 by the improper fraction \( \frac{31}{28} \):\[ 7 \times \frac{31}{28} = \frac{7 \times 31}{28} = \frac{217}{28} \]

Step 3: Simplify the Fraction

We need to simplify the fraction \( \frac{217}{28} \). First, check if 217 and 28 have any common factors. The greatest common divisor (GCD) of 217 and 28 is 1, as 217 is a prime number. Hence, the fraction is already in its simplest form.

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. A proper fraction is where the numerator (the top number) is smaller than the denominator (the bottom number). Mixed numbers are often used in everyday life, such as when telling time or measuring ingredients in cooking. However, for mathematical operations like multiplication and division, it is usually more convenient to convert a mixed number into an improper fraction.

For example, in the problem you had, the mixed number was \(1 \frac{3}{28}\). To convert this into an improper fraction, you multiply the whole number by the denominator, and then add the numerator. So for \(1 \frac{3}{28}\):

• Multiply \(1\) (the whole number) by \(28\) (the denominator), which equals \(28\).
• Add \(3\) (the numerator), resulting in \(31\).
• Place \(31\) over \(28\) to get the improper fraction \(\frac{31}{28}\).

This makes it easier to perform multiplication with other numbers.

Improper Fractions Explained

An improper fraction is where the numerator is larger than or equal to the denominator. For instance, in the fraction \(\frac{31}{28}\), the numerator \(31\) is larger than the denominator \(28\). Improper fractions are useful because they simplify mathematical operations, like multiplication, making expressions easier to calculate straight away.

You will often encounter improper fractions when you convert mixed numbers for multiplication or division. In our example, after converting \(1 \frac{3}{28}\) to \(\frac{31}{28}\), it becomes straightforward to multiply it by the whole number. It's this process of using improper fractions that helps in maintaining consistency in fraction operations.

In summary:

• Improper fractions have numerators larger than denominators.
• They are formed by converting mixed numbers.
• They simplify calculations involving fractions.

Understanding how to work with improper fractions will help you tackle a wide variety of math problems with confidence.

Simplifying Fractions

After performing operations with fractions, like multiplication, the resulting fraction can often be simplified. Simplifying a fraction involves reducing it to its smallest form without changing its value. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this GCD.

In the example of the fraction \(\frac{217}{28}\), we check for common factors. The GCD of \(217\) and \(28\) is \(1\). This tells us that the fraction is already in its simplest form, as \(217\) is a prime number, which means it only has \(1\) and itself as divisors.

Steps to simplify fractions include:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• If the GCD is \(1\), the fraction is already in simplest form.

Simplifying makes fractions easier to interpret, ensuring that you work with the most compact expression possible for clarity’s sake.