Problem 8 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. A number divided by eight, plus seven, is fifty.

Step-by-Step Solution

Answer

The equation is \( \frac{x}{8} + 7 = 50 \).

Step 1: Identify the Unknown

The problem mentions 'a number,' which we can represent with a variable, often denoted as \( x \). This will be our unknown.

Step 2: Translate the Division Phrase

The phrase 'a number divided by eight' can be translated to a mathematical expression: \( \frac{x}{8} \).

Step 3: Translate the Addition Phrase

The phrase 'plus seven' indicates that we need to add 7 to the previous expression. Thus, we get \( \frac{x}{8} + 7 \).

Step 4: Translate the Equation

The word 'is' in mathematics typically denotes equality. So the phrase 'is fifty' translates to \( = 50 \).

Step 5: Construct the Full Equation

Combine all parts to form the equation: \( \frac{x}{8} + 7 = 50 \).

Key Concepts

Mathematical Expressions

Mathematical expressions are like sentences in math, using numbers, operations, and sometimes variables to convey an idea. In the exercise provided, the phrase we want to translate is 'a number divided by eight, plus seven'. Here, we're looking at operations like division and addition.

To break it down:

• "A number divided by eight" requires us to take this unknown number (which we call a variable) and divide it by 8. In math, this is written as \( \frac{x}{8} \) where \( x \) is the unknown number.
• "Plus seven" means you add 7 to your current value, which we express as \( \frac{x}{8} + 7 \).

These smaller expressions are essential building blocks for more complex equations, helping us turn a word problem into a solvable math problem.

Variables in Math

In math, a variable is a symbol, usually a letter, that represents an unknown value. This is crucial when translating sentences into mathematical terms because it helps capture the unknown without needing to know its actual value.

In the exercise, the phrase 'a number' refers to something we don't yet know, so we use a variable — in this case, \( x \). This variable can stand for any number until further information, often found by solving an equation, reveals its true value.

Variables are powerful tools in algebra:

• They provide a way to express relationships in a consistent form.
• Help simplify and solve equations where the value isn't obvious right away.

Understanding and using variables allows for a much broader range of problems to be tackled, as they can express things not fit neatly into a static number.

Equations

Equations are mathematical statements that express equality. In our exercise, we translated the phrase 'a number divided by eight, plus seven, is fifty' into an equation. Understanding how these are structured is key to solving problems where relationships are defined.

Here's how equations work:

• Each part of a sentence gets transformed into a mathematical term or operation.
• The word 'is' joins these parts with an equal sign \( = \), asserting that both sides have the same value.

For our problem, the complete sentence translated to the equation \( \frac{x}{8} + 7 = 50 \). This says if you take 'a number' divide it by eight, add seven, you'll have exactly fifty.

Knowing how to translate complex word problems into these neat, compact forms allows us to solve for the variable using algebraic techniques, ultimately revealing unknown values.