Problem 44 - Introductory Algebra for College Students

Question

A. Rewrite the division as multiplication involving a multiplicative inverse. B. Use the multiplication from part (a) to find the given quotient. $$-18 \div 6$$

Step-by-Step Solution

Answer

-3

Step 1: Rewrite Division as Multiplication With Inverse

To rewrite \( -18 \div 6 \) as a multiplication with inverse, we simply flip the divisor (6) to get its inverse and change the operation from division to multiplication. So we get \( -18 \times \frac{1}{6} \). We can keep the negative sign with the 18 as it is.

Step 2: Perform Multiplication

Now we perform the multiplication operation with \( -18 \) and \( \frac{1}{6} \). For this operation, we simply multiply the numerator -18 with the numerator 1 and the denominator 1 with the denominator 6. This results in \( \frac{-18}{6} \).

Step 3: Simplify Fraction

The result \(\frac{-18}{6}\) needs to be simplified. By dividing the numerator and the denominator by the common factor, which is 6, we get \(-3\) as the final answer.

Key Concepts

Multiplicative Inverse

A multiplicative inverse (also called a reciprocal) is a number that, when multiplied by a given number, results in the identity, which is 1. In simpler terms, if you have a number like 6, its multiplicative inverse would be \( \frac{1}{6} \). When you multiply these two numbers together, \( 6 \times \frac{1}{6} = 1 \). This concept comes in very handy in division problems because division can be transformed into multiplication by using the multiplicative inverse.

For example, in the expression \( -18 \div 6 \), you can replace the division sign by multiplying by the inverse of 6, which is \( \frac{1}{6} \). The equation then becomes: \( -18 \times \frac{1}{6} \). Remember, for any non-zero number 'a', the multiplicative inverse is \( \frac{1}{a} \).

This approach simplifies calculations and is often used to solve problems more easily in algebra and arithmetic. Being comfortable with finding and using multiplicative inverses is key for tackling complex mathematical problems.

Multiplication

Multiplication of fractions involves multiplying the numerators together and the denominators together. When you have an expression like \( -18 \times \frac{1}{6} \), you can think of -18 as \( \frac{-18}{1} \), making the multiplication straightforward.

Here's the step-by-step breakdown:

• Multiply the numerators: \(-18 \times 1 = -18\).
• Multiply the denominators: \(1 \times 6 = 6\).
• Combine: You get the result \( \frac{-18}{6} \).

Multiplication serves as an essential tool in converting division problems into a simpler format that often leads to easier solutions. Remember, handling signs is crucial. When a negative number is involved in multiplication, such as \( -18 \times \frac{1}{6} \), the negative sign carries through to the result. This ensures consistent and accurate results when working with real numbers.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its smallest possible form. You achieve that by finding the greatest common factor (GCF) for both the numerator and the denominator and dividing them by this factor.

In the example of \( \frac{-18}{6} \), the GCF of 18 and 6 is 6. By dividing both the numerator and the denominator by their GCF, the fraction simplifies:

• \(-18 \div 6 = -3\) for the numerator.
• \(6 \div 6 = 1\) for the denominator.
• This gives \( \frac{-3}{1} \), which simplifies further to just \(-3\).

Simplifying is a crucial final step to ensure your answer is in its simplest form possible. Doing so helps to display results clearly and ensures comparative connections with other mathematical results or measurements are more apparent. Practicing this step enhances your ability to handle and interpret fractions efficiently.