- Identify the ratio between the length (\(L\)) and the number of stitches (\(S\)) from the given data points, which is consistently \(\frac{1}{5} = 0.2\).
- Recognize that a constant ratio indicates a direct proportional relationship defined by the formula \(L = kS\).
- Substitute the constant \(k = 0.2\) into the formula to obtain the relationship equation.
- The final relationship is \(\boxed{L = \frac{1}{5}S}\).
Explanation
- Analyzing the Data
The problem asks for the relationship between the number of stitches (\(S\)) and the length of a row in centimeters (\(L\)). We are provided with four data points: (5, 1), (15, 3), (20, 4), and (30, 6). By examining these points, we can determine if there is a linear or proportional relationship between the number of stitches and the length. We test the ratio \(\frac{L}{S}\) for each pair:
- For \((5, 1)\): \(\frac{1}{5} = 0.2\)
- For \((15, 3)\): \(\frac{3}{15} = 0.2\)
- For \((20, 4)\): \(\frac{4}{20} = 0.2\)
- For \((30, 6)\): \(\frac{6}{30} = 0.2\)
The ratio of the length to the number of stitches is consistently \(0.2\).
- Formulating the Relationship
Since the ratio \(\frac{L}{S}\) is a constant \(k = 0.2\), we can conclude that the relationship is directly proportional. A direct proportional relationship between two variables \(x\) and \(y\) is defined by the equation \(y = kx\), where \(k\) is the constant of proportionality. Substituting our variables and the constant value, we get \(L = 0.2S\). This can also be written in fractional form as \(L = \frac{1}{5}S\). This means that for every 5 stitches, the row length increases by 1 centimeter.
- Conclusion and Final Answer
The mathematical relationship between the number of stitches \(S\) and the length \(L\) is given by the linear equation \(L = 0.2S\) or \(\boxed{L = \frac{1}{5}S}\).
Examples
In real-life scenarios, proportional relationships like this are essential for planning and estimation. For example, if you are planning to sew a curtain of a specific length, you can use this relationship to determine exactly how many stitches you need to complete the project without having to measure repeatedly. By understanding that \(L = 0.2S\), you can quickly calculate that a row of 50 cm would require \(S = \frac{L}{0.2} = \frac{50}{0.2} = 250\) stitches. Similarly, for other tasks like cooking (scaling ingredients), construction (scaling materials), or map reading (scaling distances), identifying these consistent ratios allows for precise planning and efficient resource management.