Problem 81 - Elementary Algebra Within Reach

Question

In Exercises 79-84, evaluate the expression. $$ \left(-\frac{4}{3}\right)\left(-\frac{9}{16}\right) $$

Step-by-Step Solution

Answer

The result of the expression \((- \frac{4}{3}) \times (- \frac{9}{16})\) is \(+ \frac{3}{4}\).

Step 1: Understanding the expression

In the given expression \(-\frac{4}{3} \times -\frac{9}{16}\), the \(-\) sign denote that the fractions are negative. The task is to evaluate the multiplication of these two negative fractions.

Step 2: Multiply the numerators and denominators

To multiply fractions, the numerators of the fractions are multiplied together to get the numerator of the product, and the denominators are multiplied together to get the denominator of the product. So, \((-\frac{4}{3}) \times (-\frac{9}{16})\) becomes \(\frac{-4 \times -9}{3 \times 16} = \frac{36}{48}\).

Step 3: Simplify the resulting fraction

The resulting fraction \(\frac{36}{48}\) can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 12. The fraction thus becomes \(\frac{3}{4}\)

Step 4: Determine the sign

As a rule, the product of two negative numbers is a positive number. Hence, \((-)\times (-) = +\), and the final answer is positive: \(+\frac{3}{4}\)

Key Concepts

Negative Fraction Multiplication

When it comes to negative fraction multiplication, it may seem a bit bewildering at first, but the principle is straightforward. Multiplication of negative fractions follows the same rules as multiplying positive fractions—multiply the numerators together and the denominators together. However, the key point is remembering that a negative times a negative yields a positive result.

For instance, consider \( (-\frac{4}{3}) \times (-\frac{9}{16}) \). When you multiply the numerators, \( -4 \times -9 \), you get 36, which is positive because both numbers were negative. Similarly, multiply the denominators \( 3 \times 16 \) to get 48. Thus, you end with a positive product, \( \frac{36}{48} \), which you can then simplify further.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that involves reducing a fraction to its simplest form, where the numerator and denominator have no common divisors other than 1. To achieve this, we need to find the Greatest Common Divisor (GCD) of the numerator and denominator, and then divide them both by it.

Let's simplify \( \frac{36}{48} \). We need to find the GCD of 36 and 48. It's often helpful to list out the factors to determine the GCD. For 36, the factors are 1, 2, 3, 4, 6, 9, 12, 18, 36, and for 48, they are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The greatest common factor is 12. Divide both the numerator and the denominator by 12, which results in \( \frac{36 \div 12}{48 \div 12} = \frac{3}{4} \).

Simplification makes further calculations easier and the results more understandable, as dealing with reduced fractions such as \( \frac{3}{4} \) is much more straightforward than working with \( \frac{36}{48} \).

Greatest Common Divisor

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the highest number that divides exactly into two or more numbers without leaving a remainder. Finding the GCD is a vital step in various mathematical processes, such as simplifying fractions, adding or subtracting fractions with different denominators, and solving ratio problems.

To determine the GCD of two numbers, like 36 and 48, one could use different methods. One method is to list all factors of each number and identify the largest common one - in this case, 12. Another method is the Euclidean algorithm, which uses division to find the GCD efficiently, especially when dealing with large numbers.

Understanding and being able to compute the GCD allows students to simplify and compare fractions effectively, making it a cornerstone concept in arithmetic and algebra.