Problem 30 - Chemistry Matter and Change

Question

Apply Concepts A classmate converts 68 \(\mathrm{km}\) to meters and gets 0.068 \(\mathrm{m}\) as the answer. Explain why this answer is incorrect, and identify the likely source of the error.

Step-by-Step Solution

Answer

The classmate mistakenly divided by 1000 instead of multiplying, leading to 0.068 m instead of 68000 m.

Step 1: Understand the Unit Conversion

The conversion factor between kilometers and meters is crucial to understand. There are 1000 meters in 1 kilometer. Therefore, to convert kilometers to meters, we must multiply the number of kilometers by 1000.

Step 2: Apply the Conversion Factor

To convert 68 kilometers to meters, we use the conversion factor from the previous step. We multiply 68 by 1000 to get the number of meters: \[ 68 \text{ km} \times 1000 \frac{\text{m}}{\text{km}} = 68000 \text{ m} \]

Step 3: Analyze the Error

The error in the conversion was in the direction of multiplication. Instead of multiplying by 1000, which is the correct conversion method, the classmate likely misapplied the conversion by falsely dividing by 1000, resulting in 0.068 m instead of 68000 m.

Key Concepts

Kilometers to Meters

When we talk about converting kilometers to meters, we are essentially transforming one unit of measurement into another. Kilometers and meters are both units used to measure distance or length in the metric system, but they differ in scale. One kilometer is equivalent to 1000 meters. This means that to convert a distance from kilometers to meters, we need to change the larger unit (kilometers) into the smaller unit (meters).

• 1 Kilometer = 1000 Meters
• Kilometers are larger units compared to meters.
• Converting kilometers to meters involves multiplication.

When performing this conversion, remember we're shifting from a larger unit to a smaller unit, which makes the numerical value larger. This characteristic of units in the metric system makes conversions straightforward when you know the relationships among units.

Conversion Factor

A conversion factor is a number used to change one set of units to another, by multiplying or dividing. In the context of kilometers and meters, the conversion factor is 1000. This factor is used because there are 1000 meters in one kilometer. Using the correct conversion factor ensures accurate calculation of transformations from one measurement unit to another.

• Conversion Factor for Kilometers to Meters: 1000
• This is because 1 km equals 1000 meters.
• The factor must be used to scale the distance correctly.

Accurately using conversion factors is crucial to avoid mistakes in calculations, as seen in the exercise where a division instead of a multiplication led to a significant error.

Multiplication Error

A multiplication error occurs when a calculation involves incorrect arithmetic operations, leading to the wrong result. In the exercise, instead of multiplying 68 kilometers by 1000 to convert it into meters, an incorrect operation was applied resulting in 0.068 meters.

• A mistake might arise from misunderstanding the operation needed.
• Know that larger to smaller unit conversions require multiplication.
• Make sure to apply the correct conversion factor appropriately.

The error likely stemmed from confusion, leading to a division instead of the needed multiplication by 1000. Being aware of the rules of unit conversion and the right mathematical operations are essential to avoid errors like these.