Amanda takes 14 hours to travel 720 km, if she travels 240 km by tow truck and the rest by van. If she travels 320 km by tow truck and the rest by van, she takes 40 minutes more. Find the ratio of the speeds of the tow truck and the van.?

1. A

   5 ∶ 3

2. B

   3 :2

3. C

   2 ∶ 3

4. D

   2 ∶ 5

Appeared in: UP CHO - 10 September 2021

Explanation

• The problem requires setting up two equations based on the relationship Time = Distance / Speed for two different travel scenarios.
• In the first scenario, traveling 240 km by tow truck (speed x) and 480 km by van (speed y) takes 14 hours, leading to the equation (240/x) + (480/y) = 14.
• In the second scenario, traveling 320 km by tow truck and 400 km by van takes 14 hours and 40 minutes (44/3 hours), leading to the equation (320/x) + (400/y) = 44/3.
• Solving this system of equations yields the speed of the tow truck (x) as 40 km/h and the speed of the van (y) as 60 km/h.
• The ratio of the tow truck's speed to the van's speed is 40:60, which simplifies to 2:3.

Why Other Options Were Wrong

• Option A: This ratio (5:3) would imply the tow truck is significantly faster than the van, which contradicts the calculated speeds (40 km/h and 60 km/h).
• Option B: This is the inverse of the correct ratio (3:2 vs 2:3). It incorrectly represents the ratio of the van's speed to the tow truck's speed (60:40).
• Option D: This ratio (2:5) does not match the calculated speeds of 40 km/h and 60 km/h.

Related Visual

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving simultaneous linear equations derived from speed, distance, and time word problems as background academic context rather than a clinical decision trigger.
• While this is a mathematics question, the underlying skill of setting up and solving systems of equations is crucial in nursing for various calculations, such as drug dosages, IV drip rates, and interpreting physiological models.
• Nurses must be precise with calculations, as a mistake can have serious consequences. For example, miscalculating a ratio for a medication mixture could lead to an overdose or underdose.
• What if the second journey took 40 minutes less instead of more? The total time for the second journey would be 14 - 2/3 = 40/3 hours. The second equation would become (320/x) + (400/y) = 40/3, which would lead to a different set of speeds and a different final ratio.

How to Approach the Question

• First, identify the unknown quantities and assign variables to them. Here, the unknowns are the speeds of the tow truck (x) and the van (y).
• Read the problem carefully to identify the two distinct scenarios that will form your two equations. Note all the given values for distance and time.
• Use the fundamental formula Time = Distance / Speed to translate the information from each scenario into a mathematical equation.
• Ensure all units are consistent. In this case, convert the time given in minutes (40 minutes) into hours (2/3 hours) to match the other time unit.
• You will now have a system of two linear equations with two variables. Use a method like substitution or elimination to solve for one variable.
• Substitute the value of the first variable back into one of the original equations to find the value of the second variable.

Concept Tested & Keywords

• Concept Tested: Solving simultaneous linear equations derived from speed, distance, and time word problems.
• Stem keywords: speed, distance, time, ratio, tow truck, van
• Lead-in keywords: Find the ratio

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