Write a complete mathematical justification

A complete mathematical justification states a clear claim, represents relevant quantities or relationships, and connects calculations, diagrams, tables, or equations through logically valid steps using definitions and known properties. At this level, reasoning supports conclusions about whole numbers, fractions, decimals, ratios, and familiar geometric relationships; a single example illustrates but does not establish a general claim, and an answer without an explanation is incomplete. Formal proof, abstract symbolic generalization, and advanced edge-case analysis are not included.

Detailed Explanation: Write a complete mathematical justification

A complete mathematical justification does more than give an answer. It includes:

A clear claim—what you think is true.
Representations—numbers, equations, a table, or a diagram that show the important information.
Logical steps—an explanation of why each step is valid.
A conclusion—a sentence that connects the work back to the question.

Example

Problem: Which fraction is greater, $\frac{3}{4}$ or $\frac{5}{8}$ ? Justify your answer.

Step 1: State a claim.

I claim that $\frac{3}{4}$ is greater than $\frac{5}{8}$ .

Step 2: Represent the fractions with the same denominator.

To compare fractions, it helps to use equal-sized parts. The denominators $4$ and $8$ can both be represented using eighths.

Since $4 \times 2=8$ , multiply the numerator and denominator of $\frac{3}{4}$ by $2$ :

\frac{3}{4}=\frac{3\times 2}{4\times 2}=\frac{6}{8}.

This does not change the value of the fraction because the numerator and denominator were multiplied by the same number.

Step 3: Compare the equivalent fractions.

Now compare $\frac{6}{8}$ and $\frac{5}{8}$ . They have the same denominator, so compare their numerators:

6>5,

which means

\frac{6}{8}>\frac{5}{8}.

Because $\frac{3}{4}=\frac{6}{8}$ , it follows that

\boxed{\frac{3}{4}>\frac{5}{8}}.

Therefore, $\frac{3}{4}$ is greater than $\frac{5}{8}$ because it represents $6$ eighths, while $\frac{5}{8}$ represents only $5$ eighths.

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