Problem 67 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. Three-fourths of a number, increased by twelve

Step-by-Step Solution

Answer

The expression is \( \frac{3}{4}x + 12 \).

Step 1: Identify the components of the phrase

The phrase "Three-fourths of a number" indicates a fraction of a number. The fraction is \( \frac{3}{4} \) and the number is represented by \( x \). So, the expression for "Three-fourths of a number" is \( \frac{3}{4}x \).

Step 2: Express the phrase with addition

The phrase "increased by twelve" means to add 12 to the expression we have from Step 1. This gives us the expression \( \frac{3}{4}x + 12 \).

Step 3: Simplification

Since there are no like terms to combine in the expression \( \frac{3}{4}x + 12 \), it is already in its simplest form. Thus, no further simplification is possible.

Key Concepts

Understanding Fractions

Fractions are an essential part of algebraic expressions. They represent a part of a whole, and are especially useful when dealing with expressions involving parts of quantities, and can also represent division. In this context, the fraction \( \frac{3}{4} \) represents three parts out of four. When applied to a variable or number, like \( x \), it signifies multiplying \( x \) by \( \frac{3}{4} \).
In real-life scenarios, if you were to have three out of four slices of a pizza, you are utilizing a fraction to depict the part of the pizza you have eaten. Similarly, algebra uses fractions to manipulate variables and numbers effectively. For example, "three-fourths of a number" translates to \( \frac{3}{4}x \), indicating the operation of multiplying \( x \) by \( \frac{3}{4} \). Understanding how to work with fractions is crucial as they frequently appear in algebra.

Simplification in Algebra

Simplification aims to make an algebraic expression as straightforward as possible, without changing its value. It involves combining like terms, using arithmetic, and sometimes factoring. The main goal is to have the expression look neat and be easy to work with.

In our example, the expression \( \frac{3}{4}x + 12 \) is already in its simplest form. There are no like terms to combine since one part involves the variable \( x \) and the other is a constant (12). Simplification steps might vary depending on the complexity of the expression, but some key considerations remain:

• Combine like terms if possible, like variables with variables, and constants with constants.
• Look for and simplify any complex fractions.
• Factorise where applicable to make expressions more manageable.

Practice makes perfect when it comes to simplifying expressions, as familiarity with different types of algebraic components can help you identify simplification opportunities more quickly.

Basics of Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It is the language of mathematics that allows us to create generalized formulas and solve equations involving unknown values. Often, letters like \( x \) are used to stand for numbers we don't yet know.

In the exercise, "a number" is represented by \( x \), an unknown or variable. This variable can be manipulated through operations such as addition, subtraction, multiplication, and division. Practicing algebra helps in developing a logical approach to solving problems and understanding relationships between quantities.
Some basic elements to remember about algebra include:

• Expressions: A combination of variables, numbers, and operations.
• Equations: Statements that express the equality of two expressions, often containing one or more unknowns.
• Use of Variables: Letters or symbols that represent unknown numbers and can change values.

Mastering these fundamentals provides a solid foundation for more advanced mathematical concepts.