Problem 101 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -1 \frac{1}{4}\left(-\frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The result is \(\frac{15}{16}\).

Step 1: Convert Mixed Number to Improper Fraction

First, we need to convert the mixed number \(-1 \frac{1}{4}\) to an improper fraction. The whole part is -1, which we multiply by 4 (the denominator) and then add the numerator (1), giving us \(-1 \times 4 + 1 = -5\), so \(-1 \frac{1}{4} = -\frac{5}{4}\).

Step 2: Multiply the Fractions

Next, take the improper fraction \(-\frac{5}{4}\) and multiply it by the fraction \(-\frac{3}{4}\). The multiplication of fractions involves multiplying the numerators together and the denominators together: \[ (-5) \times (-3) = 15 \] and \[ 4 \times 4 = 16 \].

Step 3: Simplify the Resulting Fraction

The product of the fractions is \(\frac{15}{16}\). Since 15 and 16 have no common factors other than 1, the fraction \(\frac{15}{16}\) is already in its simplest form.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. This means that the fraction represents a value equal to or greater than 1. An improper fraction is useful in fraction multiplication because it simplifies calculations.

To convert a mixed number like \(-1 \frac{1}{4}\) into an improper fraction, follow these steps:

• Multiply the whole number by the denominator (in this case, \(-1\times 4 = -4\)).
• Add the numerator to this product (\(-4 + 1 = -5\)).
• Now, the mixed number \(-1 \frac{1}{4}\) transforms into the improper fraction \(-\frac{5}{4}\).

Understanding improper fractions allows you to perform further operations seamlessly and aids in the multiplication process.

Mixed Numbers

Mixed numbers consist of a whole number and a proper fraction. They are often used to express amounts greater than one in everyday situations, such as "one and a quarter". When working with mathematical operations like multiplication, it's helpful to convert mixed numbers to improper fractions first.

The reason for this conversion is that improper fractions simplify the process of multiplying, dividing, or comparing fractions. When numbers are in the form of a simple fraction, it's easier to handle calculations without managing both a whole number and a fraction separately. Always remember:

• Convert mixed numbers to improper fractions before performing operations.
• Use the result of this conversion for any subsequent actions.

Becoming comfortable with this conversion is key to solving many fraction-related problems.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. This makes the fraction easier to read and use in further calculations.

For example, after multiplying two fractions, you may end up with a compound fraction like \(\frac{15}{16}\). You should check:

• If both the numerator and denominator have a common factor, divide them by it to simplify.
• If no common factors exist (as with 15 and 16), the fraction is already simplified.

Being able to quickly simplify fractions is an important skill as it makes your results cleaner and sometimes easier to interpret. Remember, while simplifying doesn't change the value of the fraction, it does make it more user-friendly.