Problem 66 - Algebra 1

Question

Find the resulting unit of measure. (years) \(\cdot\) (people per year)

Step-by-Step Solution

Answer

The resulting unit of measure is 'people'.

Step 1: Identify the Units

The units given are 'years' and 'people per year'. These units are being multiplied together.

Step 2: Apply the Multiplication

When multiplying units, similar units will cancel each other out. In this case, 'years' from 'people per year' cancels out with 'years'. This is similar to how, in mathematics, similar terms cancel each other out in a fraction.

Step 3: Find the Resulting Unit

With 'years' cancelled out, 'people' remain as the resulting unit.

Key Concepts

Dimensional Analysis

Dimensional analysis is an essential tool in physics and engineering used to convert units from one measurement system to another and to check the physical validity of equations. It involves the process of including unit labels during calculations to ensure that the final result has the appropriate unit of measure.

For example, the exercise involves calculating the resulting unit of measure by multiplying 'years' by 'people per year'. Dimensional analysis starts with identifying the unit for each quantity and paying close attention to how these units interact when applied to mathematical operations. It's like having a conversation in a language of units where the rules of grammar must match up to make sense. This process ensures that the equations used in scientific calculations are not just numerically but dimensionally consistent as well, serving as a vital check against computational errors.

Units Cancellation

Units cancellation is a crucial component of dimensional analysis, particularly when dealing with equations involving multiplication or division of different units. The principle is based on the fact that a unit divided by itself equals one, rendering it dimensionless.

When two units that are inverses of each other are multiplied, they effectively cancel out. This is apparent in our example, where multiplying 'years' by 'people per year' leads to the 'years' units canceling each other out, much like when simplifying fractions. Visualization might help students here; imagine you have a basket of apples and you take away an equal number of apples - what you're left with no longer contains any apples. Similarly, after cancellation, we're left with the unit 'people', removing the temporal dimension from our initial compound unit.

Multiplying Units

Multiplying units involves a straightforward process, much like multiplying numerical values. However, instead of just numbers, you must deal with dimensions. To correctly multiply units, you align them such that units with the same dimension either reinforce each other if they're in the same part of a fraction, or cancel out if they are on opposing sides of a fraction.

Referring back to our example, by multiplying 'years' with 'people per year', we’re essentially dealing with a fraction where 'years' is in the numerator and denominator, thus canceling out. Remembering that units behave like variables can be a helpful tip for students, where identical units, like identical variables, can be cancelled out when they appear on both the top and bottom of a division operation.