Problem 40 - Prealgebra and Introductory Algebra

Question

The president of a company traveled 1800 mi by jet and 300 \(\mathrm{mi}\) on a prop plane. The rate of the jet was four times the rate of the prop plane. The entire trip took 5 h. Find the rate of the jet.

Step-by-Step Solution

Answer

The speed of the jet is 600 mi/h

Step 1: Set Up the Equations

Assume that the rate of the jet is \(r_{jet}\) and of the prop plane is \(r_{plane}\). From the problem, we know that \(r_{jet} = 4*r_{plane}\) and the time spent on the jet, \(t_{jet}\), plus the time spent on the prop plane, \(t_{plane}\), is equal to 5 hours. We can express time as the ratio of distance to rate, so \(t_{jet} = \frac{1800}{r_{jet}}\) and \(t_{plane} = \frac{300}{r_{plane}}\). Adding these two equations gives us \(\frac{1800}{r_{jet}} + \frac{300}{r_{plane}} = 5\)

Step 2: Solve the System of Equations

Plug the equation for the jet's rate into the equation obtained in Step 1: \(\frac{1800}{4*r_{plane}} + \frac{300}{r_{plane}} = 5\). Simplifying gives us \(\frac{450}{r_{plane}} + \frac{300}{r_{plane}} = 5\), or \(\frac{750}{r_{plane}} = 5\), which simplifies to \(r_{plane} = \frac{750}{5} = 150 \mathrm{mi/h}\)

Step 3: Derive the Speed of the Jet

Finally, substitute \(r_{plane} = 150 \mathrm{mi/h}\) into the equation for the jet's rate, \(r_{jet} = 4*r_{plane}\), yielding \(r_{jet} = 4*150 = 600 \mathrm{mi/h}\)

Key Concepts

Distance Formula

When tackling rate problems, the distance formula, \[ distance = rate \times time \] plays a crucial role. This formula allows us to calculate any one of the components (distance, rate, or time) if the other two are known. In the context of the problem provided, we break down the journey into two parts: jet travel and prop plane travel.
The objective here is to derive how long each segment took. For jet travel, we know the distance was 1800 miles and the formula modifies to\[ t_{jet} = \frac{1800}{r_{jet}} \] where \( t_{jet} \) is the time for jet travel, and \( r_{jet} \) is the rate of the jet. Similarly, for the prop plane, we use\[ t_{plane} = \frac{300}{r_{plane}} \] where \( t_{plane} \) refers to the time for prop plane travel and \( r_{plane} \) is its rate.
These relationships are essential as they set up the stage for forming a system of equations that incorporates all known data.

System of Equations

A system of equations is a set of two or more equations with the same variables. Solving these systems provides a powerful method for finding unknowns in word problems such as the one at hand. In our exercise, we've derived two relationships:

• \( r_{jet} = 4 \times r_{plane} \)
• \( \frac{1800}{r_{jet}} + \frac{300}{r_{plane}} = 5 \)

By substituting the expression for \( r_{jet} \) into the second equation, we effectively reduce the number of variables, simplifying the problem to one variable, \( r_{plane} \). This step helps us solve for \( r_{plane} \) first, which can then be used to find \( r_{jet} \).
Understanding how to manipulate and solve systems of equations is fundamental in solving rate problems where multiple modes of travel or interconnected speeds are involved.

Problem-Solving Strategies

Approaching problems logically with effective strategies simplifies complex scenarios. Here are a few steps involved in tackling these types of exercises:

Identify the Known and Unknown

Begin by pinpointing what you know and what needs to be determined. In this case, we knew the distances and the relationship between the two rates.

Formulate Equations

Use the known values to write expressions. This might include creating relationships between variables and establishing expressions for time as seen in our solution.

Solve Step-by-Step

Proceed methodically to resolve the equations using algebraic manipulation. Start with substitutions that simplify to solve one variable first and proceed to others.
Each of these strategies is about making the problem more manageable by breaking it down into smaller parts. By prioritizing logical organization, you can bring clarity and method to your problem-solving approach.