Problem 115 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x y\) for the given values of \(x\) and \(y.\) $$x=1 \frac{3}{13}, y=-6 \frac{1}{2}$$

Step-by-Step Solution

Answer

The evaluated expression \(x y\) is \(-88/13)

Step 1: Convert the Mixed Fractions into Improper Fractions

Firstly, convert the mixed number \(x = 1 \frac{3}{13}\) into an improper fraction. Multiply the denominator of the fractional part (13) by the whole number (1) and add the numerator of the fractional part (3) to get the new numerator. Keep the denominator the same. Do the same for \(y\). So the improper fractions are: \[x = \frac{(1*13) + 3}{13} = \frac{16}{13}\] \[y = \frac{(-6*2) + 1}{2} = \frac{-11}{2}\]

Step 2: Substitute and Multiply

Now you're going to substitute these values into the expression \(x y\) and perform the multiplication:\[x y = \frac{16}{13} * \frac{-11}{2}\] Multiply the numerators together to get the numerator of the result and the denominators together to get the denominator of the result.

Step 3: Simplify

On multiplication you get: \[x y = \frac{16 * -11}{13 * 2} = \frac{-176}{26}\] Now simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. \[x y = \frac{-88}{13}\]

Key Concepts

Mixed Fractions to Improper Fractions

When working with mixed numbers, such as the given values of x and y in the exercise, it is often necessary to convert them to improper fractions to simplify the calculation process. A mixed fraction consists of a whole number and a proper fraction. To convert a mixed fraction, like x = 1 \frac{3}{13}, into an improper fraction, follow this simple procedure:
Multiply the denominator of the fraction by the whole number, and then add the numerator of the fraction. The result becomes the numerator of the new improper fraction, while the denominator remains unchanged.
For instance:
\[x = 1 \frac{3}{13} = \frac{(1 \times 13) + 3}{13} = \frac{16}{13}\]
This method provides a straightforward approach to dealing with mixed fractions in any algebraic expression.

Multiplication of Fractions

Multiplying fractions might seem daunting, but it's based on a straightforward rule. To multiply any two fractions, simply multiply their numerators (top numbers) together to find the new numerator, and multiply their denominators (bottom numbers) together to find the new denominator.
So, using our previous example where x has been converted to an improper fraction \( \frac{16}{13} \) and y to \( \frac{-11}{2} \) as:\[x \times y = \frac{16}{13} \times \frac{-11}{2}\]
We multiply the numerators and denominators respectively:
\[= \frac{16 \times -11}{13 \times 2}\]
This will yield the product of the two fractions in the form of another fraction.

Simplifying Fractions

The final step in evaluating the given expression involves simplifying the fraction obtained from multiplications. To simplify a fraction, we need to find the Greatest Common Divisor (GCD) of the numerator and denominator and then divide both by that number.
For the product of x and y, we have the fraction \( \frac{-176}{26} \) from the multiplication step. The GCD of 176 and 26 is 2. So to simplify:
\[= \frac{-176 \div 2}{26 \div 2}\]
\[= \frac{-88}{13}\]
Remember, the objective of simplification is to make the fraction as simple as possible, ideally until no further division by common factors can occur. The result is a simplified fraction representing the same quantity or value.