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Bunuel
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Bunuel
Official Solution:

A steamer takes 4 hours and 30 minutes to travel from Town A to Town B when traveling upstream against the current. However, it only takes the steamer 3 hours to travel from Town B to Town A when traveling downstream with the current. Assuming the current flows at a constant speed, how long will it take a raft, floating downstream at the speed of the current, to travel from Town B to Town A ?

A. 10 hours
B. 12 hours
C. 15 hours
D. 18 hours
E. 20 hours


Assume the speed of the steamer in still water is \(s\) and the speed of the current to is \(c\).

When traveling upstream against the current, the steamer's speed is \(s - c\), and in 4.5 hours it covers a distance of \(4.5(s - c)\).

When traveling downstream with the current, the steamer's speed is \(s + c\), and in 3 hours it covers a distance of \(3(s + c)\).

Since both distances represent the distance between Towns A and B, we can equate them: \(4.5(s - c)=3(s + c)\), which gives \(s = 5c\). To express the distance in terms of only one unknown, substitute \(s = 5c\) in either of the above equations, to get \(3(s + c)=18c\).

To determine how long a raft, floating downstream at the speed of the current, will take to travel from Town B to Town A, we divide the distance between the towns by the speed of the raft: \(\frac{distance}{rate}=\frac{18c}{c}=18\) hours.


Answer: D
So is the effective speed of raft now 2c? ( speed of current + speed of raft which also equals speed of current)?
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Bunuel
Official Solution:

A steamer takes 4 hours and 30 minutes to travel from Town A to Town B when traveling upstream against the current. However, it only takes the steamer 3 hours to travel from Town B to Town A when traveling downstream with the current. Assuming the current flows at a constant speed, how long will it take a raft, floating downstream at the speed of the current, to travel from Town B to Town A ?

A. 10 hours
B. 12 hours
C. 15 hours
D. 18 hours
E. 20 hours


Assume the speed of the steamer in still water is \(s\) and the speed of the current to is \(c\).

When traveling upstream against the current, the steamer's speed is \(s - c\), and in 4.5 hours it covers a distance of \(4.5(s - c)\).

When traveling downstream with the current, the steamer's speed is \(s + c\), and in 3 hours it covers a distance of \(3(s + c)\).

Since both distances represent the distance between Towns A and B, we can equate them: \(4.5(s - c)=3(s + c)\), which gives \(s = 5c\). To express the distance in terms of only one unknown, substitute \(s = 5c\) in either of the above equations, to get \(3(s + c)=18c\).

To determine how long a raft, floating downstream at the speed of the current, will take to travel from Town B to Town A, we divide the distance between the towns by the speed of the raft: \(\frac{distance}{rate}=\frac{18c}{c}=18\) hours.


Answer: D
So is the effective speed of raft now 2c? ( speed of current + speed of raft which also equals speed of current)?

No, the effective speed of the raft is not 2c. The raft floats at the same speed as the current, which is just c. Since it has no additional speed of its own, it simply moves at the current's speed, c.
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Hi Bunuel,

I am not 100% clear on why the speed of the raft is c and not 2c, isnt it that the effective raft speed will become 2c downstream? Could you explain please?

Thank you so much!
Bunuel
Sukritvarsh
Bunuel
Official Solution:

A steamer takes 4 hours and 30 minutes to travel from Town A to Town B when traveling upstream against the current. However, it only takes the steamer 3 hours to travel from Town B to Town A when traveling downstream with the current. Assuming the current flows at a constant speed, how long will it take a raft, floating downstream at the speed of the current, to travel from Town B to Town A ?

A. 10 hours
B. 12 hours
C. 15 hours
D. 18 hours
E. 20 hours


Assume the speed of the steamer in still water is \(s\) and the speed of the current to is \(c\).

When traveling upstream against the current, the steamer's speed is \(s - c\), and in 4.5 hours it covers a distance of \(4.5(s - c)\).

When traveling downstream with the current, the steamer's speed is \(s + c\), and in 3 hours it covers a distance of \(3(s + c)\).

Since both distances represent the distance between Towns A and B, we can equate them: \(4.5(s - c)=3(s + c)\), which gives \(s = 5c\). To express the distance in terms of only one unknown, substitute \(s = 5c\) in either of the above equations, to get \(3(s + c)=18c\).

To determine how long a raft, floating downstream at the speed of the current, will take to travel from Town B to Town A, we divide the distance between the towns by the speed of the raft: \(\frac{distance}{rate}=\frac{18c}{c}=18\) hours.


Answer: D
So is the effective speed of raft now 2c? ( speed of current + speed of raft which also equals speed of current)?

No, the effective speed of the raft is not 2c. The raft floats at the same speed as the current, which is just c. Since it has no additional speed of its own, it simply moves at the current's speed, c.
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Hi Bunuel,

I am not 100% clear on why the speed of the raft is c and not 2c, isnt it that the effective raft speed will become 2c downstream? Could you explain please?

Thank you so much!
Bunuel
Sukritvarsh
So is the effective speed of raft now 2c? ( speed of current + speed of raft which also equals speed of current)?

No, the effective speed of the raft is not 2c. The raft floats at the same speed as the current, which is just c. Since it has no additional speed of its own, it simply moves at the current's speed, c.

The key point is that the raft does not have its own propulsion; it moves solely due to the flow of the current. Therefore, its speed is equal to the speed of the current, c.

If the raft had its own propulsion, you could potentially consider adding speeds, but since it simply floats with the current, it cannot go faster than the current itself. Thus, the effective speed of the raft downstream is c, not "c + c" or "2c."

This concept distinguishes between objects with their own propulsion (like the steamer) and those that rely entirely on the current (like the raft).
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I don’t quite agree with the solution. It is assumed that speed of raft itself is 0. Poor question framing.
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Shivam2024
I don’t quite agree with the solution. It is assumed that speed of raft itself is 0. Poor question framing.
A raft is a simple, flat, floating structure made of materials like logs or planks, often without any engine, motor, or sails. It doesn't generate its own movement. So when placed in a river, it just drifts with the flow of the water, that is, it moves at the speed of the current.

So, in general a raft has no independent speed. Its motion depends entirely on external forces, like a river current. That’s why, in this question, the raft's speed is taken as equal to the current’s speed.
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I like the solution - it’s helpful.
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