Question

Write the verbal sentence as an equation. One fourth of a number is 36

Short answer

The equation is \( \frac{x}{4} = 36 \).

Step-by-step solution

Identify the unknown

Let's denote the number we're trying to find as x. This is the unknown value in our equation.

Translate 'one fourth of a number'

The phrase 'one fourth of a number' means we multiply our unknown x by 1/4. This can be written as \( \frac{1}{4} * x \) or \( \frac{x}{4} \). It shows the fraction of the number we're working with.

Translate 'is 36'

When we see 'is' in a verbal sentence, we can interpret it as the equality sign (=) in our equation. The '36' becomes the other side of the equation.

Write the full equation

Combining all parts together, the verbal sentence translates into the following equation: \( \frac{x}{4} = 36 \)

Key concepts

Writing Equations

Translating verbal sentences into mathematical equations is a foundational skill in algebra. It involves recognizing the relationships and operations described in words and representing them with mathematical symbols. In our example, the phrase 'One fourth of a number is 36' can be interpreted as an equation by following specific steps. First, you identify the components of the phrase: 'one fourth' indicates a fraction and 'is' suggests equality. Together, they form the equation \( \frac{x}{4} = 36 \). Breaking down sentences into smaller parts and understanding the meaning of each word within the mathematical context is vital for writing accurate equations.

When practicing, remember to:

• Identify the operation (addition, subtraction, multiplication, division)
• Look for words that signal equality such as 'is', 'equals' or 'results to'
• Understand how to represent fractions or ratios

By mastering these steps, writing equations will become a clear translation of the problem at hand rather than a mere guess.

Unknown Variables

Algebraic equations often contain unknowns, which are typically represented by letters such as \(x\), \(y\), or \(z\). These unknown variables stand in for numbers we wish to find. In our exercise, the unknown is simply 'a number' which we have denoted as \(x\). It's important to choose a variable that makes sense in the context of the problem.

When dealing with unknowns, keep in mind to:

• Clearly define what the variable represents
• Consistently use the same variable throughout the problem
• Understand that the variable is a placeholder for a single, specific value

Grasping the concept of unknown variables allows you to translate verbal problems into solvable equations effectively.

Fractions in Algebra

Fractions are a crucial part of algebra, especially when it comes to representing parts of a whole or ratios. In algebraic terms, a fraction like \(\frac{1}{4}\) is understood as one part of four equal parts.

In our example, the verb phrase 'one fourth of a number' prompts us to multiply the variable by \(\frac{1}{4}\). In the equation, this is represented as either \(\frac{1}{4} * x\) or \(\frac{x}{4}\).

Fraction Multiplication

When you multiply a number by a fraction, you are essentially scaling down the number by that fraction.

Operational Symbols

Remember, multiplication can be represented by a dot (\(\cdot\)) or by simply placing the variable next to the fraction (\(\frac{1}{4}x\)).

Fractions in Equations

To solve equations involving fractions, you may need to perform operations such as cross-multiplication or find a common denominator, which are essential techniques for manipulating equations with fractions.

Equality in Mathematics

Equality in mathematics signifies that two expressions have the same value. In the language of algebra, the equals sign \(=\) is used to express this relationship.

Understanding equality is pivotal for solving equations because it maintains that actions performed on one side of the equation must also be applied to the other to keep the equation balanced.

Balance Concept

Think of an equation as a balance scale, where both sides need to have the same weight for the scale to be balanced.

Evaluating Both Sides

When you see a phrase like 'is 36' in a verbal sentence, this means that the expression on one side of the equality is equivalent to 36 on the other side. Therefore, in our problem, \(\frac{x}{4}\) 'is' equated to 36, forming the equation \(\frac{x}{4} = 36\).

This concept extends beyond simple numbers to complex expressions and ensures each step taken during the solving process maintains the original equality.