- Identify the independent variable as the input (Amount of Rice) and the dependent variable as the output (Amount of Water).
- Recognize that the amount of water required varies directly as a function of the amount of rice, represented by the linear equation \(W = 2.5 \times R\).
- Conclude that the correct statements are:
- The amount of rice is the independent value.
- The amount of water is the dependent value.
- \(\boxed{\text{The amount of rice is the independent value; The amount of water is the dependent value.}}\)
Explanation
- Analyzing the Data Relationship
To determine the relationship between the amount of rice and the amount of water, we examine the data in the table. The table provides pairs of values: \((2, 5)\), \((3, 7.5)\), \((5, 12.5)\), and \((8, 20)\). In any functional relationship represented by a table, the first column typically represents the independent variable, and the second column represents the dependent variable. In this scenario, we choose the amount of rice to cook, and the amount of water required to cook that rice is then determined. Therefore, the amount of rice is the variable we control (the input), and the amount of water is the variable that responds to that control (the output).
- Identifying Independent and Dependent Variables
An independent variable is a variable that is changed or controlled in a scientific experiment or mathematical model to test the effects on a dependent variable. A dependent variable is a variable being tested and measured, and it changes based on the changes made to the independent variable. Based on our analysis:
- The 'Amount of Rice (cups)' is the independent value because it is the chosen input.
- The 'Amount of Water (cups)' is the dependent value because its amount depends on the quantity of rice being prepared.
By checking the ratio \(\frac{\text{Water}}{\text{Rice}}\), we see \(\frac{5}{2} = 2.5\), \(\frac{7.5}{3} = 2.5\), \(\frac{12.5}{5} = 2.5\), and \(\frac{20}{8} = 2.5\). This confirms the linear functional relationship \(W = 2.5 \times R\), where \(W\) is the amount of water and \(R\) is the amount of rice. Because \(W\) is calculated from \(R\), it is clearly the dependent variable.
- Selecting the Correct Options
Comparing our findings with the provided options:
- 'The amount of rice is the dependent value.' (Incorrect)
- 'The amount of water is the dependent value.' (Correct)
- 'The amount of rice is the independent value.' (Correct)
- 'The amount of water is the independent value.' (Incorrect)
- 'The values cannot be labeled as dependent or independent without a given equation.' (Incorrect)
The two correct statements are that the amount of rice is the independent value and the amount of water is the dependent value.
Examples
Understanding the difference between independent and dependent variables is essential in daily life, such as when grocery shopping. For instance, the total cost of apples is a dependent variable, while the number of pounds of apples you decide to purchase is the independent variable. The price per pound is the constant of proportionality that links them: \(\text{Total Cost} = \text{Price per pound} \times \text{Weight}\). Just like the rice-water ratio, this relationship helps you predict the outcome of your actions before you reach the checkout counter.