- To find the first product, compute \(6 \times \frac{2}{3} = 4\). Since \(\frac{2}{3} < 1\), the result is less than 6.
- To find the second product, convert \(1 \frac{1}{2}\) to \(\frac{3}{2}\) and compute \(6 \times \frac{3}{2} = 9\). Since \(\frac{3}{2} > 1\), the result is greater than 6.
- The calculated products are \(4\) and \(9\).
- Final answer: \(\boxed{4, 9}\)
Explanation
- Problem Analysis and Objectives
We aim to compute the products \(6 \times \frac{2}{3}\) and \(6 \times 1 \frac{1}{2}\). We also need to analyze why the first product is smaller than \(6\) and why the second is larger than \(6\). Recall that when we multiply a positive integer \(n\) by a rational number \(r\), the product \(n \times r\) is smaller than \(n\) if \(r < 1\), and the product is larger than \(n\) if \(r > 1\).
- Calculating and Analyzing the First Product
First, let us calculate \(6 \times \frac{2}{3}\). Multiplying the integer by the numerator and dividing by the denominator gives us:
\[6 \times \frac{2}{3} = \frac{6 \times 2}{3} = \frac{12}{3} = 4.\]
Since \(\frac{2}{3}\) is less than \(1\), multiplying \(6\) by this fraction means we are taking only a portion of the original number. Because \(0 < \frac{2}{3} < 1\), it follows logically that \(6 \times \frac{2}{3} < 6 \times 1\), resulting in \(4 < 6\).
- Calculating and Analyzing the Second Product
Next, we calculate \(6 \times 1 \frac{1}{2}\). First, we convert the mixed number to an improper fraction:
\[1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}.\]
Now, we calculate the product:
\[6 \times \frac{3}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9.\]
Because \(1 \frac{1}{2}\) is greater than \(1\), we are essentially multiplying \(6\) by a value larger than one, which naturally results in a product greater than the initial number. Thus, \(9 > 6\).
Examples
Understanding how multiplying by fractions changes a value is essential in daily life. For instance, if you are baking and a recipe calls for 6 cups of flour but you only want to make two-thirds of the recipe, you multiply \(6 \times \frac{2}{3} = 4\) cups. Conversely, if you want to make one-and-a-half times the recipe, you multiply \(6 \times 1.5 = 9\) cups. Recognizing that multiplying by a fraction less than 1 decreases the total and multiplying by a number greater than 1 increases it helps in adjusting quantities, planning projects, and managing resources effectively.