Problem 11 - Introductory Algebra for College Students

Question

perform the indicated multiplication. $$\frac{1}{2}(-24)$$

Step-by-Step Solution

Answer

The solution to the problem \( \frac{1}{2}(-24) \) is -12.

Step 1: Interpret the Negative sign

Understand that the negative sign associated with the integer (-24) means that the number is less than zero and the result of the multiplication will also be negative if the other number is positive.

Step 2: Multiply the fraction with the integer

Proceed with the multiplication operation by multiplying the fraction 1/2 with 24. Multiplication of a fraction with an integer can be done by multiplying the integer with the numerator of the fraction (in this case the numerator is 1), and keeping the denominator as it is. Hence, in this case \( \frac{1}{2} \times 24 = \frac{24}{2} \).

Step 3: Simplify the fraction

By division, simplify the fraction \( \frac{24}{2} \) which equals 12.

Step 4: Include the negative sign

Now, don't forget to include the negative sign from the integer (-24) to get the final answer, which will be -12.

Key Concepts

Fraction Multiplication

Multiplying fractions can be simple once you understand the process. Start by taking a look at the components of a fraction: the numerator (the top number) and the denominator (the bottom number). When you multiply a fraction by an integer, you only need to multiply the numerator of the fraction by the integer. For instance, with the fraction \( \frac{1}{2} \) and the integer \( -24 \), focus first on multiplying the integer with just the numerator \( 1 \), which results in \( 24 \).

• The denominator always stays the same, so it remains as \( 2 \) in this instance.
• After multiplying, you will end up with \( \frac{24}{2} \).

This setup allows you to simplify easily afterwards.

Simplifying Fractions

Once you've performed the multiplication in your fraction, you almost always want to simplify it to make it easier to interpret. In our example with \( \frac{24}{2} \), simplifying involves dividing the numerator by the denominator. This means performing the division \( 24 \div 2 \), which equals \( 12 \).

• After division, the fraction reduces from \( \frac{24}{2} \) to just \( 12 \), making it a whole number which is easier to read.
• Sometimes fractions might not divide evenly, and you'll need to use the greatest common divisor (GCD) to find the simplest form.

Simplifying is a significant step because a simplified fraction or number is much quicker to work with.

Integer Multiplication

Multiplying with integers becomes straightforward with regular practice. Remember the rules concerning negative and positive signs. Multiply the numbers regularly, then apply the sign rule:

• A positive number times a positive number equals a positive result.
• A negative number times a positive number equals a negative result.
• A negative number times a negative number equals a positive result.
• So in this case, multiplying \( -24 \) with \( \frac{1}{2} \), the result remains negative, leading to \( -12 \).

Knowing these rules helps ensure accuracy and confidence when multiplying integers, regardless of whether they are positive or negative.