Problem 115 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The sum of 15 and \(-3,\) divided by the product of 4 and -3

Step-by-Step Solution

Answer

-1

Step 1 - Interpret the phrase

First, identify the numerical values and operations described in the phrase. The phrase contains two parts: 'The sum of 15 and \(-3\)' and 'divided by the product of 4 and -3'.

Step 2 - Write the expression

Convert the words into a mathematical expression. The sum of 15 and -3 is written as \(15 + (-3)\). The product of 4 and -3 is written as \(4 \times (-3)\). So, the full expression is \((15 + (-3))/(4 \times (-3))\).

Step 3 - Simplify the numerator

Simplify the numerator: \(15 + (-3) = 15 - 3 = 12\)

Step 4 - Simplify the denominator

Simplify the denominator: \(4 \times (-3) = -12\)

Step 5 - Simplify the fraction

Now, divide the simplified numerator by the simplified denominator: \(12 / (-12) = -1\)

Key Concepts

Numerical Expression

A numerical expression uses numbers and operations to represent a specific value or set of values. In our exercise, we need to convert the phrase into a mathematical form, often involving addition, subtraction, multiplication, and division. The process involves:

• Identifying numerical values mentioned in the phrase (e.g., 15, -3, 4).
• Determining the operations to perform on these values (e.g., sum and product).
• Writing the expression using appropriate mathematical symbols.

For the given phrase, 'the sum of 15 and -3, divided by the product of 4 and -3,' the corresponding numerical expression is \(\frac{{15 + (-3)}}{{4 \times (-3)}}\).

Simplification

Simplification means reducing an expression to its simplest form. This involves performing arithmetic operations and combining like terms. Let's break down the simplification process for our example:

• First, simplify the numerator: \(15 + (-3) = 12\). Adding a negative number is the same as subtracting the absolute value of that number, so \(15 - 3 = 12\).
• Next, simplify the denominator: \(4 \times (-3) = -12\). Multiplying a positive number by a negative number results in a negative product.
• Finally, simplify the entire fraction: \(\frac{{12}}{{-12}} \). Since dividing a positive number by a negative number results in a negative quotient, the simplified result is \ -1 \.

Simplifying ensures the expression is easier to understand and use in further calculations.

Division

Division is the process of finding how many times one number is contained within another. It involves a dividend (the number to be divided) and a divisor (the number by which we divide). Here are some key points:

• When dividing two numbers with the same sign (both positive or both negative), the quotient is positive.
• When dividing numbers with different signs (one positive, one negative), the quotient is negative.

In our exercise, the numerator is 12, and the denominator is -12. Using the rule of dividing numbers with different signs, \(\frac{{12}}{{-12}} = -1\). This basic understanding of division helps us perform and verify calculations accurately.

Negative Numbers

Negative numbers are values less than zero and are symbolized by a minus sign (-). They obey specific rules in arithmetic operations:

• Adding a negative number is equivalent to subtracting its positive counterpart. For example, \(15 + (-3) = 15 - 3\).
• Multiplying or dividing a negative number by a positive number results in a negative number. For instance, \(4 \times (-3) = -12\) and \(\frac{12}{-12} = -1\).
• Multiplying or dividing two negative numbers yields a positive number (e.g., \(-3) \times (-4) = 12\).

Understanding the behavior of negative numbers simplifies their handling in various mathematical contexts, ensuring accurate and reliable results.