Problem 6 In each of Exercises 1–10, mat... [FREE SOLUTION] (2024)

Short Answer

Percentages

Percentages are a way to express a number as a fraction of 100. They are denoted using the percent symbol \(%\). For example, 50% is the same as saying 50 out of 100 or 0.50. When trying to find out what percent one number is of another, we use a simple formula:
\[ \text{Percentage} = \frac{\text{part}}{\text{whole}} \times 100 \]
Here, the 'part' is the smaller value or the value we are focused on, and the 'whole' is the larger value or the total amount. This formula helps simplify percentage problems, whether in a classroom setting or everyday life.
For instance, if you need to find what percent 23 is of 57, identify 23 as the part and 57 as the whole. Then apply the formula to get the percentage. This straightforward method makes understanding percentages much simpler.

Equation Setup

Setting up the correct equation is crucial for solving algebra word problems. In percentage problems, you often translate a word problem into a math equation. This involves identifying the variables and constants involved and structuring them correctly.
For the problem '23 is what percent of 57?', we start by setting up an equation using the percentage formula. Knowing that:
\[ \text{Percentage} = \frac{23}{57} \times 100 \]
This equation allows us to identify the variables and isolate what we need to solve for. In this case, we substitute 23 as the 'part' and 57 as the 'whole'. This setup forms the basis for solving the equation. It ensures accuracy and helps lay a solid foundation for finding the solution.
Correctly setting up equations is essential in algebra as it directly leads to finding the right answers efficiently and accurately.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In percentage word problems, these expressions are used to represent the parts of our problem numerically and symbolically.
Take the exercise '23 is what percent of 57?' After setting up the equation, our algebraic expression is:
\[ 23 = 0.57 \times y \]
In general, for any percentage word problem, you may re-write it as:
\[ \text{part} = \left( \frac{\text{part}}{\text{whole}} \right) \times 100 \]
Using this algebraic expression, you can easily manipulate and solve for the variable you're interested in, such as the percentage (\(y\)) in this case. Understanding and being able to craft algebraic expressions is key in translating word problems into solvable equations, which is a fundamental skill in algebra.