A, B, K start from the same place and travel in the same direction at speeds of 30km/hr, 40km/hr, 60km/hr...

1. Translate the problem requirements: Three people A, B, K travel at different speeds (\(30\), \(40\), \(60\) km/hr) from the same starting point. B starts \(2\) hours after A. We need to find how many hours after A that K started, given that B and K both overtake A at the exact same moment.
2. Set up the overtaking condition: For B and K to overtake A at the same instant, all three must be at the same location at that moment, meaning they've all traveled the same distance.
3. Express distances in terms of time: Use the time when overtaking occurs as our reference point and work backwards to determine when each person started based on \(\mathrm{distance} = \mathrm{speed} \times \mathrm{time}\).
4. Solve for K's start time: Set up equations where distances are equal and solve for the unknown start time of K.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English: We have three people - A, B, and K - all starting from the same place and heading in the same direction, but at different speeds and starting times.

A travels at \(30\) km/hr and starts first (let's call this time \(0\)).
B travels at \(40\) km/hr and starts \(2\) hours after A.
K travels at \(60\) km/hr and starts at some unknown time after A (this is what we need to find).

The key condition is that B and K both catch up to (overtake) A at exactly the same moment. This means all three people are at the same location at that instant.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Set up the overtaking condition

For B and K to overtake A at the same instant, think of it this way: imagine you're watching from above and see all three people meet at the same spot on the road at the same time. This means they've all traveled the exact same distance from the starting point, even though they started at different times and traveled at different speeds.

Since \(\mathrm{distance} = \mathrm{speed} \times \mathrm{time}\), and all distances are equal at the moment of overtaking:
Distance traveled by A = Distance traveled by B = Distance traveled by K

3. Express distances in terms of time

Let's say the overtaking happens \(\mathrm{t}\) hours after A started.

At this moment:

• A has been traveling for \(\mathrm{t}\) hours at \(30\) km/hr
• B has been traveling for \((\mathrm{t} - 2)\) hours at \(40\) km/hr (since B started \(2\) hours late)
• K has been traveling for \((\mathrm{t} - \mathrm{x})\) hours at \(60\) km/hr, where \(\mathrm{x}\) is what we want to find

So our distances are:

• A's distance: \(30\mathrm{t}\)
• B's distance: \(40(\mathrm{t} - 2)\)
• K's distance: \(60(\mathrm{t} - \mathrm{x})\)

4. Solve for K's start time

Since all distances are equal at the overtaking moment:

First, let's find when A and B meet:
\(30\mathrm{t} = 40(\mathrm{t} - 2)\)
\(30\mathrm{t} = 40\mathrm{t} - 80\)
\(-10\mathrm{t} = -80\)
\(\mathrm{t} = 8\) hours

So the overtaking happens \(8\) hours after A started.

Now, since K also meets them at this same moment:
\(30\mathrm{t} = 60(\mathrm{t} - \mathrm{x})\)
\(30(8) = 60(8 - \mathrm{x})\)
\(240 = 480 - 60\mathrm{x}\)
\(60\mathrm{x} = 480 - 240\)
\(60\mathrm{x} = 240\)
\(\mathrm{x} = 4\)

Therefore, K started \(4\) hours after A.

Process Skill: MANIPULATE - Systematically solving the system of equations

5. Final Answer

K started \(4\) hours after A.

Verification: At \(\mathrm{t} = 8\) hours after A started:

• A has traveled: \(30 \times 8 = 240\) km
• B has traveled: \(40 \times (8 - 2) = 40 \times 6 = 240\) km
• K has traveled: \(60 \times (8 - 4) = 60 \times 4 = 240\) km

All three have traveled the same distance, confirming our answer.

The answer is C. \(4\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "overtake" as different arrival times

Students often misunderstand that B and K "overtake A at the same instant" means all three are at the same location at the same time. They might think it means B and K each separately catch up to A at different times, leading them to set up two separate problems instead of one unified condition where all distances are equal.

2. Confusion about reference time frame

Students frequently struggle with choosing a consistent time reference point. They might measure B's time from when B starts (instead of from when A starts) or measure K's time from when K starts, creating inconsistent equations. The key insight is that all times should be measured from when A starts (\(\mathrm{t} = 0\)).

3. Incorrectly setting up the constraint equations

Students may set up speed equations instead of distance equations, or they might incorrectly assume that the time differences between start times should be equal (thinking K starts \(2\) hours after B, making it \(4\) hours total). The correct approach requires understanding that we need equal distances at the moment of overtaking.

Errors while executing the approach

1. Arithmetic errors in solving linear equations

When solving \(30\mathrm{t} = 40(\mathrm{t} - 2)\), students often make sign errors or distribution mistakes. For example, they might get \(30\mathrm{t} = 40\mathrm{t} + 80\) instead of \(30\mathrm{t} = 40\mathrm{t} - 80\), or incorrectly solve \(-10\mathrm{t} = -80\) to get \(\mathrm{t} = -8\) instead of \(\mathrm{t} = 8\).

2. Forgetting to substitute the found value of t

After finding \(\mathrm{t} = 8\) hours from the first equation (A and B meeting), students sometimes forget to use this value in the second equation with K. They might try to solve the K equation independently without using the constraint that all three meet at the same time.

3. Calculation errors in the final substitution

When solving \(30(8) = 60(8 - \mathrm{x})\), students often make errors in the arithmetic: calculating \(30 \times 8 = 320\) instead of \(240\), or incorrectly manipulating \(240 = 480 - 60\mathrm{x}\) to get wrong values for \(\mathrm{x}\).

Errors while selecting the answer

1. Selecting the meeting time instead of K's start time

Students might correctly calculate that the overtaking occurs \(8\) hours after A starts, but then select this as their final answer instead of recognizing that the question asks for how many hours after A did K start (which is \(\mathrm{x} = 4\)).

2. Providing the time from when K starts rather than from when A starts

Some students might calculate correctly but then give their answer as "K travels for \(4\) hours before meeting" rather than "K starts \(4\) hours after A started," potentially selecting a wrong answer choice if they misread the options.