As a logistics planner, you are analyzing two delivery routes. A freight train and a cargo truck both travel a distance of 300 miles to a distribution center. The train travels 15 mph faster than the truck, which has an average speed of $r$ mph. You must model the scenario where the train takes 2 hours less than the truck to arrive. Which equation correctly recalls the distance, rate, and time problem-solving strategy to translate this scenario?

In a uniform motion application involving a time difference, such as comparing a bike ride to a bus ride over the same distance, the relationship between time ($t$), distance ($d$), and rate ($r$) is defined by the formula $t = \frac{d}{r}$.

As a logistics coordinator for a regional delivery service, you are analyzing two different transportation methods for a 120-mile route. One method is 15 mph faster than the other and takes less time to arrive. To find the speeds, you must apply the standard problem-solving strategy for uniform motion. Arrange the following steps in the correct order to solve this type of application.

As a dispatcher for a regional delivery service, you are setting up a travel-time comparison between a standard delivery van and an express truck for a 75-mile route. The van travels at an average speed of $r$ miles per hour, and the express truck travels 10 miles per hour faster. Match each component of this logistics scenario with its correct algebraic expression.

Algebraic Travel Time Modeling in Logistics

As a logistics analyst for a regional courier service, you are modeling the travel times of a local bicycle courier and a delivery van over a 75-mile route. The van travels 10 miles per hour faster than the courier ($r$), and completes the journey in 2 hours less time. To find the speeds, you set up the following rational equation:

$\frac{75}{r} - 2 = \frac{75}{r + 10}$

To solve this equation for $r$, your first step is to multiply both sides by the least common denominator (LCD) to clear the fractions. Recall the algebraic expression for this LCD.

The least common denominator (expressed in factored form) is ____.

Standard Operating Procedure for Uniform Motion Setup