Problem 80 - Algebra and Trigonometry

Question

Accuracy of a Scale \(\quad\) A coffee merchant sells a customer 3 lb of Hawaiian Kona at \(\$ 6.50\) per pound. His scale is accurate to within \(\pm 0.03\) lb. By how much could the cus- tomer have been overcharged or undercharged because of possible inaccuracy in the scale?

Step-by-Step Solution

Answer

The customer could be overcharged or undercharged by about $0.20.

Step 1: Understand the Problem

The customer buys coffee at a rate of $6.50 per pound. However, the scale used for weighing is not perfectly accurate. It can deviate by plus or minus 0.03 lb from the declared weight of 3 lb. This means the actual weight could be between 2.97 lb and 3.03 lb. We need to find out how much this deviation could cause an overcharge or undercharge.

Step 2: Calculate Potential Undercharging

If the scale weighs the coffee at 2.97 lb instead of the intended 3 lb, the merchant sells 0.03 lb less coffee. To find the potential undercharge, calculate the monetary difference for this weight discrepancy: the price of 0.03 lb of coffee is \(0.03 \times 6.50 = 0.195\). The customer could be undercharged by $0.195 (or about 20 cents).

Step 3: Calculate Potential Overcharging

If the scale weighs the coffee at 3.03 lb instead of 3 lb, the merchant sells 0.03 lb more coffee. To find the potential overcharge, calculate the monetary difference for this weight discrepancy: the price of 0.03 lb of coffee is \(0.03 \times 6.50 = 0.195\). The customer could be overcharged by $0.195 (or about 20 cents).

Step 4: Summarize the Effects

The scale’s inaccuracy may cause the customer to be undercharged by up to $0.195 if too little coffee is weighed, or overcharged by up to $0.195 if too much coffee is weighed.

Key Concepts

Scale Accuracy

Scale accuracy is crucial when selling goods by weight, such as coffee. In this context, the scale's accuracy refers to how precisely it measures the actual weight against the displayed weight. An accurate scale should display the true weight, but mechanical or digital scales may have a margin of error. This margin is often defined by a plus-minus value indicating possible deviation from the indicated weight.
In the given problem, the scale is accurate to within ±0.03 lbs. This means the actual quantity of coffee could be anywhere from 2.97 lbs to 3.03 lbs when it says 3 lbs.
Understanding this small deviation helps in analyzing potential discrepancies in transactions involving weighted goods. It's important to regularly check and calibrate scales to minimize errors, thus maintaining trust and fairness in commercial exchanges.

Weight Deviation Calculation

Weight deviation calculation involves determining the range of possible weight differences due to scale inaccuracies. This is critical in understanding how much the actual weight might differ from the measured weight.
For instance, if you are buying 3 lbs of coffee, and the scale's accuracy allows a deviation of ±0.03 lbs, you should calculate the extremes of the weight range:

• Minimum possible weight = 3 lbs - 0.03 lbs = 2.97 lbs
• Maximum possible weight = 3 lbs + 0.03 lbs = 3.03 lbs

These calculations tell us that the weight could legally vary within this range without necessarily implying incorrect measurement. Knowing these limits is useful for estimating the potential impact on pricing and customer satisfaction.

Financial Impact Analysis

Financial impact analysis is used to understand the monetary consequences of weight inaccuracies. Even small deviations can result in either overcharging or undercharging, affecting both the merchant and the customer financially.
To analyze the financial impact in our problem, calculate how the 0.03 lbs deviation affects the cost:

• If the weight is underestimated by 0.03 lbs, the customer is undercharged by approximately \( 0.03 \times 6.50 = \$0.195 \) or 19.5 cents.
• If the weight is overestimated by 0.03 lbs, the customer is overcharged similarly by \( 0.195 \).

Understanding this variance highlights the importance of using precise and well-maintained scales to avoid financial discrepancies and maintain fair trade practices. By examining these financial variances, sellers can make informed decisions about scale maintenance and customer communication to ensure smooth operations.