Problem 100 - Elementary Algebra

Question

Perform the operations. Reduce answers to lowest terms. $$ 12 \cdot(-45) \div 1415 $$

Step-by-Step Solution

Answer

The result is \(-\frac{108}{283}\).

Step 1: Perform Multiplication

First, multiply the numbers given: \(12\) and \(-45\). The result is computed by multiplying the coefficients and considering the signs. Thus, \(12 \times (-45) = -540\).

Step 2: Perform Division

Next, take the result from Step 1, which is \(-540\), and divide it by \(1415\). We perform the division: \(-540 \div 1415\). This yields \(-\frac{540}{1415}\).

Step 3: Reduce to Lowest Terms

To simplify \(-\frac{540}{1415}\), we need to find the greatest common divisor (GCD) of \(540\) and \(1415\). The GCD of \(540\) and \(1415\) is 5. Dividing both numerator and the denominator by their GCD gives \(-\frac{540 \div 5}{1415 \div 5} = -\frac{108}{283}\). "-\frac{108}{283}" is in its lowest terms as \(108\) and \(283\) have no common divisor other than 1.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations used in mathematics, involving the product of two numbers. In our exercise, we have the numbers \(12\) and \(-45\). When multiplying these, you follow these steps:

• Multiply the absolute values: \(12 \times 45 = 540\).
• Apply the sign rules: A positive number times a negative number results in a negative number.

Thus, multiplying \(12\) by \(-45\) yields \(-540\). This part of the problem shows you how essential it is to understand how multiplication handles signs while calculating the product.

Division

Division is used to find out how many times one number is contained within another. After finding the product of the multiplication step to be \(-540\), the next step is to divide it by \(1415\). Division involves the following steps:

• Write down the numbers as a fraction: \(-\frac{540}{1415}\).
• Consider the division of signs: as both the numerator and the denominator are positive, the result maintains the sign from the multiplication, staying negative.

This division results in a fraction that is not currently in its simplest form. Continuing the process includes simplifying this fraction.

Greatest Common Divisor

Simplifying fractions involves finding a number that is the largest divisor common to both the numerator and denominator, called the Greatest Common Divisor (GCD). Our fraction, \(-\frac{540}{1415}\), can be simplified:

• The GCD of \(540\) and \(1415\) is determined to be \(5\).
• Division of both the numerator and denominator by the GCD gives: \(-\frac{108}{283}\).

To find the GCD, you can list out the factors of each number and find the largest one they have in common. Alternatively, use the Euclidean algorithm: a method of successively dividing and taking remainders, simplifying the process.

Negative Numbers

Working with negative numbers is an essential skill in mathematics. Negative numbers can alter the result's sign in operations like multiplication or division. When calculating with \(12\) and \(-45\), the result becomes negative due to the rule:

• A positive number multiplied or divided by a negative number results in a negative result.

Understanding this ensures accuracy when dealing with arithmetic operations. Remember, if the signs match (both numbers positive or negative), the result is positive. If they differ, the outcome is negative.