- Translate "three times a number" as the product of 3 and the variable \(x\), yielding the expression \(3x\).
- Interpret the diagram of three \(x\) boxes as the sum \(x + x + x\), which is equivalent to \(3x\).
- Identify that the correct representation is the pair \(3x \implies\) □ \(x\) □ \(x\) □ \(x\).
- Final Answer: \(\boxed{3x \implies \square x \square x \square x}\)
Explanation
- Translating the phrase into an expression
To represent the phrase "three times a number" algebraically, we identify the key components. "A number" is represented by a variable, let's call it \(x\). The phrase "three times" denotes multiplication of that variable by 3. Therefore, the algebraic expression for "three times a number" is \(3 \times x\) or simply \(3x\).
- Interpreting the diagram
A visual representation can help clarify the meaning of multiplication. When we have an expression like \(3x\), it represents the sum of the variable added to itself three times, which is \(x + x + x\). A diagram consisting of three boxes, each containing an \(x\), visually depicts this addition: □ \(x\) □ \(x\) □ \(x\). This matches the definition of multiplication where \(n \cdot a = a + a + \dots + a\) (n times).
- Selecting the correct option
Comparing the options, we look for the one that correctly pairs the algebraic expression \(3x\) with the visual representation of three \(x\) units. The option marked with ⨀ is \(3x \implies\) □ \(x\) □ \(x\) □ \(x\). This is the correct representation, as it aligns the multiplication expression with the repetitive addition shown in the diagram.
Examples
This concept is fundamental in real-life budgeting and planning. If you are buying three notebooks that cost \(x\) dollars each, the total cost is "three times the price of one notebook," which is \(3x\). Similarly, if you want to set aside three equal servings of a food item, each being \(x\) grams, you need a total of \(3x\) grams. Recognizing these patterns allows us to translate word problems into equations that are much easier to solve.