Question 129·Hard·Linear Equations in One Variable
A passenger train traveling at a constant speed covers 180 miles in the same time that a freight train covers 135 miles. The passenger train travels 15 miles per hour faster than the freight train. What is the speed, in miles per hour, of the freight train?
(Express the answer as an integer)
For uniform-motion word problems, immediately translate words into the relationship . Choose a variable for the unknown speed (usually the slower object), express the other speed in terms of that variable, and write an equation based on the condition given (here, equal times). Clear fractions by cross-multiplying, solve the resulting linear equation carefully, and double-check that you answer exactly what the question asks (freight vs. passenger speed). © aniкo.ai
Hints
Identify the key relationship
The problem says both trains take the same amount of time. How can you write time in terms of distance and speed?
Express each train’s time using a variable
Let the freight train’s speed be . Then the passenger train’s speed is . Write each train’s travel time using .
Set up and solve the equation
Set the two time expressions equal and solve the resulting equation for . Use cross-multiplication to clear the fractions. Anіkо.аi - ЅАT Рrер
Desmos Guide
Enter time expressions as functions
In Desmos, let represent the freight train’s speed. Enter two functions: y = 135/x and y = 180/(x+15).
Find the intersection point
Zoom or adjust the viewing window so you can see where the two curves intersect. Use the point-of-intersection tool (tap the intersection) to get the coordinates of that point. Аniко - Freе ЅAT Prеp
Read the freight train’s speed
Look at the x-coordinate of the intersection point. That x-value is the freight train’s speed in miles per hour, because it makes the two travel times equal.
Step-by-step Explanation
Define variables and relate speeds
Let be the speed of the freight train (in miles per hour).
The passenger train travels 15 miles per hour faster, so its speed is .
Use time = distance ÷ speed and set up an equation
Time is distance divided by speed. Wrіtten bу Аnіkο
- Time for the freight train to travel 135 miles: .
- Time for the passenger train to travel 180 miles: .
We are told these times are the same, so write the equation:
Solve the equation for the freight train’s speed
Cross-multiply to clear the fractions:
Distribute on the left:
Subtract from both sides:
Isolate f and interpret the solution
Divide both sides of by 45:
So the freight train’s speed is miles per hour.