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Chapter 1: Problem 11

Multiply and divide. $$ -15(-2) \div 10(-3) $$

Short Answer

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Step by step solution

01

Multiply the Numbers

Start by multiplying the first two numbers in the expression: \(-15 \times (-2)\). Multiply the absolute values and then consider the signs. The absolute values are \(15\) and \(2\), so the product is \(30\). Since both numbers have different signs, the result is negative. Thus, \(-15 \times (-2) = 30\).
02

Divide the Result by the Other Two Numbers

Now take the result from Step 1 and divide it by the product of the next set of numbers: \(10 \times (-3)\). Again, multiply the absolute values; \(10 \times 3 = 30\). Because one number is positive and the other is negative, the product is negative: \(10 \times (-3) = -30\). Now, divide the result from Step 1 by this product: \(30 \div (-30)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Values
In mathematics, the absolute value of a number refers to its distance from zero on the number line, regardless of its direction. Simply put, it is the number without its sign. Absolute values always result in non-negative quantities.

For example, the absolute value of \(-5\) is \(5\), and the absolute value of \(5\) is also \(5\). In operations like multiplication and division, focusing on absolute values first helps streamline calculations by temporarily ignoring the signs. Once you have multiplied or divided the absolute values, you can then consider the signs to determine the result's final sign.
Sign Rules for Multiplication and Division
The sign rules for multiplication and division are crucial to solving problems involving integers. These rules help us determine the sign of the result after performing operations.

  • When multiplying or dividing two numbers with the same sign, whether both are positive or both are negative, the result is always positive. For example, \((-2) \times (-2) = 4\).
  • When the signs are different, one positive and one negative, the result is negative. For instance, \(10 \times (-3) = -30\).
Knowing these rules allows you to correctly assign a sign to your answer after calculating the absolute value.
Order of Operations
The order of operations is a set of rules that mathematicians use to solve expressions unambiguously. It鈥檚 often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This sequence ensures that calculations are performed correctly.

In the exercise provided, multiplication and division should be addressed from left to right. First, you start with the multiplication \(-15 \times (-2)\) and then proceed to the division \(30 \div (-30)\). This clear path ensures that you calculate each step accurately and reach the correct answer.

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