Problem 49 - Algebra 1: Concepts and Skills

Question

Writing the units of measure helps you determine the units for the answer. This is called unit analysis. When the same units occur in the numerator and the denominator, you can cancel them. Use unit analysis to evaluate the expression. The letter \(h\) is an abbreviation for hours, while \(m i\) stands for miles. a. \((3 h)\left(\frac{25 m i}{1 h}\right)\) b. \((90 \mathrm{mi}) \div\left(\frac{45 \mathrm{mi}}{1 \mathrm{h}}\right)\) Evaluate the expression \((80 \mathrm{mi}) \div\left(\frac{20 \mathrm{mi}}{1 \mathrm{h}}\right)\)

Step-by-Step Solution

Answer

The solutions are: a) \(75 mi\), b) \(2h\), \(4h\)

Step 1: Solve the first expression

For the first expression \((3 h)\left(\frac{25 m i}{1 h}\right)\), multiply 3 hours by the ratio \(\frac{25 mi}{1h}\). Notice how the unit hours appears in both the numerator and denominator and thereby cancel out, leaving only miles (mi) as the unit in the result: \(3h \times \frac{25 mi}{1h} = 75 mi\)

Step 2: Solve the second expression

For the second expression \((90 mi) \div\left(\frac{45 mi}{1 h}\right)\), divide 90 miles by the ratio \(\frac{45 mi}{1h}\). The unit of miles can be cancelled out as it appears in both the numerator and the denominator, leaving only hours (h) in the result: \(90 mi \div \frac{45 mi}{1h} = 2h\)

Step 3: Solve the third expression

The third expression is \((80 mi) \div\left(\frac{20 mi}{1 h}\right)\). By dividing 80 miles by the ratio \(\frac{20 mi}{1h}\), and cancelling the repeated mile unit, it gets simplified to: \(80 mi \div \frac{20 mi}{1h} = 4h\)

Key Concepts

Dimensional Analysis

Dimensional analysis, also known as unit analysis, is a critical tool in problem-solving and understanding how units of measurement relate in equations. It involves the process of including units of measure when you perform calculations. The core idea is simple: units, like numbers, can be multiplied and divided, and when the same units appear both at the top (numerator) and at the bottom (denominator) of a fraction, they can be canceled out, just as you would with numbers.

For instance, when you multiply three hours by a ratio expressed as the quantity of miles per hour, the hour units cancel each other out, demonstrating that the resultant quantity you're left with measures in miles. This is not just a mathematical trick; it reflects how we understand the relationships between different physical quantities in the real world.

Ratios and Proportions

Ratios and proportions are fundamental concepts that describe the relationship between different quantities. A ratio is essentially a comparison between two numbers and indicates how many times one value contains another. For example, in the expression \(\frac{25 mi}{1 h}\), the ratio tells us that for every one hour, 25 miles are covered. It sets up a direct comparison between miles and hours.

A proportion, on the other hand, is an equation that states that two ratios are equal. It is used to solve for unknowns in problems involving similar figures or rates, like speed. When you work through exercises involving ratios and proportions, dimensional analysis becomes particularly useful to ensure that units are consistent and logically aligned throughout the calculations.

Units of Measure

Units of measure are standard quantities used to specify physical quantities. In the provided exercises, the units of measure are hours (h) and miles (mi). Ensuring the correct units are the product of a calculation is as important as the numerical result. This is why unit analysis is employed in solving the exercises.

When various units interact through multiplication or division, they may form new derived units like miles per hour (mi/h) or cancel each other out if they are the same. Mastery of unit manipulation allows you to easily convert and compare quantities, a skill that's invaluable in many fields, from chemistry to engineering, and even in everyday life when, for example, calculating travel times or fuel efficiency.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and operations but do not have an equality sign as an equation does. The expressions in the exercise \( (3 h)\left(\frac{25 m i}{1 h}\right) \) and \( (90 mi) \div\left(\frac{45 mi}{1 h}\right) \) are examples of algebraic expressions where the use of dimensional analysis simplifies the problem.

In these expressions, we treat the variables representing units of measure (like 'h' for hours and 'mi' for miles) almost as if they were algebraic variables. You multiply and divide them following the same principles as you would with purely numerical algebraic expressions. Understanding how to manipulate these expressions is crucial for converting and calculating accurate measurements in scientific and mathematical problems.