Problem 22 - Algebra 1: Concepts and Skills

Question

Find the product. \(-12\left(-\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The product of -12 and -1/4 is 3.

Step 1: Identify the Numbers to be Multiplied

The numbers given are -12 and -1/4.

Step 2: Multiply the Numbers

Perform the multiplication -12 * -1/4.

Step 3: Determine the Sign

According to the rule of multiplication, the multiplication of two negative numbers results in a positive number. Hence, the product of -12 and -1/4 is positive.

Step 4: Write Down the Answer

The product of -12 and -1/4 is 3.

Key Concepts

Multiplication Rules

Understanding the rules of multiplication is crucial, especially when dealing with negative numbers. These rules help determine the sign of the product based on the signs of the numbers being multiplied:

• When you multiply a positive number by another positive number, the result is positive.
• A positive number multiplied by a negative number gives a negative result.
  For example, 8 multiplied by -3 equals -24.
• When both numbers are negative, the product is positive. For instance, -7 times -5 equals 35.
• A negative number multiplied by a positive number results in a negative number.
  As an example, -6 multiplied by 2 equals -12.

In the given exercise, multiplying -12 by -1/4 uses the rule that two negative numbers produce a positive result. This is why the product of -12 and -1/4 is positive 3.

Rational Numbers

A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. This includes whole numbers, fractions, and decimals that either terminate or repeat. For example:

• Whole numbers like 3 or -6 are rational because they can be written as 3/1 or -6/1.
• Fractions like 3/7 or -1/4 are rational numbers.
• Decimals such as 0.75 (which equals 3/4) or -1.333... (which is -4/3) are also rational numbers because they can be expressed as fractions.

In our exercise, -1/4 is a rational number. It is important to know this because it helps in understanding how to multiply with rational numbers effectively. When you multiply rational numbers, like -1/4, you follow the same rules as you do with integers, remembering to keep track of the signs during multiplication.

Integer Multiplication

Integer multiplication involves the multiplication of whole numbers, which can be positive or negative. Integer multiplication follows the multiplication rules, especially when negative and positive integers are involved.

To multiply integers, irrespective of whether they are positive or negative, you can follow these steps:

• Multiply the absolute values of the numbers. The absolute value is the number without considering its sign. For example, the absolute values of -7 and 3 are 7 and 3 respectively.
• Apply the rules of multiplication to determine the sign of the product:
• If both integers are positive, the result is positive.
• If one integer is positive and the other is negative, the result is negative.
• If both integers are negative, the result is positive.

In the exercise, the integer -12 is multiplied by the rational number -1/4. By ignoring the negative signs momentarily, and treating them as positive, the absolute values are 12 and 1/4. Then, the rules of multiplication can be applied to reveal the product sign, which results in a positive product, 3.