Problem 168 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{-\frac{9}{16}}{\frac{33}{40}} $$

Step-by-Step Solution

Answer

-\frac{15}{22}

Step 1 - Identify the problem

The expression to simplify is \(\frac{-\frac{9}{16}}{\frac{33}{40}}\). This is a complex fraction.

Step 2 - Rewrite as a multiplication problem

To simplify this, recall that dividing by a fraction is the same as multiplying by its reciprocal. So, rewrite \(\frac{-\frac{9}{16}}{\frac{33}{40}}\) as \(-\frac{9}{16} \times \frac{40}{33}\).

Step 3 - Multiply the fractions

Multiply the numerators together and the denominators together: \(-\frac{9 \times 40}{16 \times 33} = -\frac{360}{528}\).

Step 4 - Simplify the fraction

To simplify \(-\frac{360}{528}\), find the greatest common divisor (GCD) of 360 and 528, which is 24. Divide both the numerator and the denominator by 24: \(-\frac{360 \div 24}{528 \div 24} = -\frac{15}{22}\).

Key Concepts

Understanding Fractions

Fractions are used to represent parts of a whole. They consist of a numerator, which is the number on top, and a denominator, which is the number on the bottom. A fraction tells you how many parts of that size there are. For example, in the fraction \(-\frac{9}{16}\), -9 is the numerator and 16 is the denominator. The fraction represents -9 out of 16 parts.

• The numerator is the number above the line. It shows how many parts we have.
• The denominator is the number below the line. It shows the total number of parts.

Understanding fractions is key when simplifying complex fractions.

Reciprocals of Fractions

A reciprocal of a fraction is obtained by flipping the numerator and the denominator. The reciprocal of \(-\frac{9}{16}\) is \(-\frac{16}{9}\). Reciprocals are useful because dividing by a fraction is the same as multiplying by its reciprocal. For example, in the exercise, we rewrite the division problem as a multiplication problem:

• To find the reciprocal, just swap the numerator and the denominator.
• Instead of dividing by a fraction, multiply by its reciprocal to simplify.

In the problem, this means changing \(-\frac{9}{16} \) to \(-\frac{9}{16} \times \frac{40}{33}\), making the math simpler.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD is essential in simplifying fractions. In the problem, after multiplying, we get the fraction \(-\frac{360}{528}\). The GCD of 360 and 528 is 24.

• To find the GCD, list the factors of both numbers and find the largest common one.
• Divide both the numerator and the denominator by this GCD for the simplest form of the fraction.

Using the GCD, divide \(-360\) by 24 and \528\ by 24, resulting in \(-\frac{15}{22}\). This gives us the final simplified fraction.