Problem 20 - Intermediate Algebra

Question

Translate each statement into mathematical symbols. Do not solve. 16 is what percent of \(55 ?\)

Step-by-Step Solution

Answer

The equation is \(16 = \frac{55x}{100}\).

Step 1: Identify the unknown value

This problem asks us to determine 'what percent' 16 is of 55. This indicates that our unknown is the percentage value. We can represent this unknown percentage as a variable, say, \(x\%\).

Step 2: Set up the mathematical equation

To translate '16 is what percent of 55' into a mathematical equation, we can use the formula for percentage: \(\text{part} = \left(\frac{\text{percentage}}{100}\right) \times \text{whole}\). Here, the 'part' is 16, the 'whole' is 55, and the 'percentage' is \(x\). The equation becomes:\[16 = \left(\frac{x}{100}\right) \times 55\].

Step 3: Write the final mathematical expression

The equation from Step 2 can be simplified to set up the problem: \[16 = \frac{55x}{100}\]. This expression exactly represents the mathematical translation of the problem statement '16 is what percent of 55.'

Key Concepts

Translating Word Problems to Equations

Translating word problems into equations can seem challenging at first, but it is a helpful skill to master. When you're faced with a problem, the goal is to extract the mathematical expressions and variables hidden in the text. Take each piece of information from the problem and think about how it can be represented using numbers or symbols.
In the exercise '16 is what percent of 55?' the task involves understanding the role of each term:

• "16" is the part of the whole you are dealing with.
• "what percent" is the unknown that you're trying to find, usually represented by the variable.
• "of 55" indicates the whole value.

By identifying these components, you create a clear path towards forming an equation. The key is breaking down each element and thinking about how they connect mathematically. This way, you translate the verbal description into a mathematical equation, ready for solving.

Mathematical Expressions

Understanding mathematical expressions is crucial in forming equations. A mathematical expression is a combination of numbers, symbols, and operators that represent a value or a relationship. Expressions don't have to be solved immediately; they can simply describe a scenario, like this problem through the formula for percentage: \[\text{part} = \left(\frac{\text{percentage}}{100}\right) \times \text{whole}\]
In the exercise, '16 is what percent of 55?', the problem is expressed as:

• "part" is 16, which is a known value you have.
• The "percentage" is unknown, represented by \(x\) which makes it a variable.
• "whole" is described by the number 55, another known value.

An important part of dealing with expressions is understanding the meaning of each part and how they contribute to the overall equation. The formula not only shows the relationship but also guides you in setting up accurate calculations later on. Each expression aligns different aspects of the problem to form an equation ready to unveil the unknown.

Unknown Variables in Algebra

In algebra, working with unknown variables is a fundamental skill. Unknowns are often represented by letters such as \(x\), and they stand for quantities we need to figure out. In percentage problems, like this exercise, the unknown variable represents a percentage value.
For example, in the equation set up from the word problem:\[16 = \frac{55x}{100}\]

• The letter \(x\) represents the unknown percentage.
• Writing the equation helps visualize the problem, noting how \(x\) is embedded in the fraction that relates the part to the whole.

Approaching the problem with a variable sets a framework for finding unknown information by further manipulating the equation. Once the equation is established, it opens a structure that later calculations will follow to solve for \(x\). Understanding variables and how they interact in an equation is vital for solving numerous mathematical challenges.