Problem 20 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. nine-sixteenths of the thickness

Step-by-Step Solution

Answer

The expression is \( \frac{9}{16}t \).

Step 1: Identify the Mathematical Operation and Fraction

The phrase 'nine-sixteenths of' suggests a multiplication involving a fraction. Here, 'nine-sixteenths' is the fraction \( \frac{9}{16} \).

Step 2: Determine the Variable for Thickness

Since no specific variable is given for 'thickness,' we will choose a variable to represent it. Let's use \( t \) to represent the thickness.

Step 3: Formulate the Algebraic Expression

The phrase 'nine-sixteenths of the thickness' can now be written as \( \frac{9}{16} \times t \). In algebraic terms, this is written as \( \frac{9}{16}t \).

Key Concepts

Fraction Multiplication

Multiplying fractions in algebraic expressions is a fundamental concept that ties to various mathematical operations. When you come across a phrase like "nine-sixteenths of," it indicates the multiplication of a fraction with another value. In our example, the fraction mentioned is \( \frac{9}{16} \). This means you are supposed to multiply this fraction by whatever the value of the other quantity is, in this case, the thickness.

Fraction multiplication can be broken down into simpler steps:

• First, identify the fraction, which here is \( \frac{9}{16} \).
• Understand that 'of' in mathematical terms often signals multiplication.
• Remember that multiplying fractions involves multiplying the numerators together and the denominators together.

For example, if you multiply \( \frac{9}{16} \) by any number, say \( t \), you perform \( 9 \times t \) for the numerator, and the denominator stays 16, simply because you are multiplying by a whole number.

Variable Representation

In algebra, variables are symbols used to represent unknown quantities. They are an essential part of forming algebraic expressions. In our exercise, the term "thickness" does not have a specified variable. It provides an opportunity to assign a variable that makes the expression clearer and easier to manipulate.

Choosing an appropriate variable is crucial:

• Common practice is to select a letter that suggests the quantity it represents; for example, \( t \) for thickness, \( x \) for an unknown, or \( v \) for volume.
• Variables allow you to generalize the expression, so it can be used with any specific value later.

In the exercise, "thickness" is conveniently assigned the variable \( t \). This translates the verbal expression into an algebraic expression, making it \( \frac{9}{16}t \). Variables facilitate the ease of doing further calculations and communicating mathematical ideas succinctly.

Mathematical Operations

Understanding mathematical operations is vital when translating phrases into algebraic expressions. The operations include addition, subtraction, multiplication, and division, all essential in forming accurate mathematical representations of real-world situations.

• Keywords like "of" or "times" often imply multiplication.
• "Plus" or "sum" suggests addition, whereas "minus" indicates subtraction.
• "Per" or "divided by" is a cue for division operations.

In the given task, the phrase "nine-sixteenths of the thickness" signifies that we need to multiply the fraction \( \frac{9}{16} \) by the unknown quantity represented by \( t \). This multiplication is seamlessly transformed into the algebraic expression \( \frac{9}{16}t \). By understanding these operational cues, you can translate real-world problems into mathematical language more effectively.