- Calculate the ratio of weight (\(W\)) to dye (\(D\)) for each data point provided: \(\frac{21}{3}=7\), \(\frac{42}{9}=\frac{14}{3}\), \(\frac{84}{12}=7\), and \(\frac{105}{15}=7\).
- Identify that the ratios are \(7\), \(\frac{14}{3}\), \(7\), and \(7\).
- Recognize that \(\frac{14}{3} \approx 4.67\) is the only ratio different from \(7\).
- Conclude that the weight of clothing corresponding to this unique ratio is \(\boxed{42}\).
Explanation
- Analyzing the Data and Identifying the Goal
To identify the weight of clothing for which the ratio of weight to dye is different, we must first analyze the given data. We are provided with a table showing pairs of weight (W) in pounds and cups of dye (D):
- Point 1: \((W, D) = (21, 3)\)
- Point 2: \((W, D) = (42, 9)\)
- Point 3: \((W, D) = (84, 12)\)
- Point 4: \((W, D) = (105, 15)\)
Our goal is to calculate the ratio \(R = \frac{W}{D}\) for each data point and identify the weight \(W\) where this ratio differs from the others.
- Calculating the Ratios
We calculate the ratio of weight to dye (\(R = \frac{W}{D}\)) for each row of the table:
- For the first pair (21, 3): \(R_1 = \frac{21}{3} = 7\).
- For the second pair (42, 9): \(R_2 = \frac{42}{9}\). Simplifying this by dividing both numerator and denominator by their greatest common divisor, 3: \(R_2 = \frac{42 \div 3}{9 \div 3} = \frac{14}{3} \approx 4.67\).
- For the third pair (84, 12): \(R_3 = \frac{84}{12} = 7\).
- For the fourth pair (105, 15): \(R_4 = \frac{105}{15} = 7\).
- Comparing Ratios and Conclusion
Comparing the calculated ratios:
- \(R_1 = 7\)
- \(R_2 = \frac{14}{3} \approx 4.67\)
- \(R_3 = 7\)
- \(R_4 = 7\)
It is clear that the ratio for the second pair, where the weight is \(42\) pounds, is \(\frac{14}{3}\), while the ratios for all other weights are \(7\). Therefore, the weight for which the ratio is different is \(42\) pounds.
Examples
In real-life scenarios like manufacturing or chemical processing, consistent ratios are crucial for quality control. If a machine's calibration requires a specific ratio of a substance to a weight, any deviation from this ratio indicates a potential error or anomaly. By checking these ratios, we can quickly identify faulty data points or miscalibrated equipment. For instance, in a painting workshop, if the mixture of paint to thinner must remain consistent, verifying these ratios helps ensure the final product meets standard specifications.