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Six children are standing along the x-axis at points (0,0), (30,0), (8 14 May 2021, 04:29
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C
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95% (hard)

Question Stats:

46% (02:56) correct 54% (02:52) wrong based on 187 sessions

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Six children are standing along the x-axis at points (0,0), (30,0), (87,0), (142,0), (237,0), (504,0). The children decide to meet at some point along the x-axis. What is the minimum total distance the children must walk in order to meet?

A. 504
B. 652
C. 766
D. 880
E. 1016
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C
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Six children are standing along the x-axis at points (0,0), (30,0), (8 14 May 2021, 06:10
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nick1816
Six children are standing along the x-axis at points (0,0), (30,0), (87,0), (142,0), (237,0), (504,0). The children decide to meet at some point along the x-axis. What is the minimum total distance the children must walk in order to meet?

A. 504
B. 652
C. 766
D. 880
E. 1016
Show more


So the students are at
0....30.....87.....142.....237.......504

We have to ensure that there are least number of students moving in same direction.
A point such that 2 moving towards east and 4 moving towards west will involve more travelling as compared to a point that involves 3 on each extreme moving towards each other.

So basically we are looking at all moving towards the median distance, which is \(\frac{87+142}{2}=114.5\)

Distance travelled = \((114.5-0)+(114.5-30)+(114.5-87)+(142-114.5)+(237-114.5)+(504-114.5)\)

\(114.5-0+114.5-30+114.5-87+142-114.5+237-114.5+504-114.5\)

\(504+237+142-87-30-0=883-117=766\)

The travelling will be the same even when you have all travelling towards any of the middle students. That is the one at 87 or the one at 147.


C
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Six children are standing along the x-axis at points (0,0), (30,0), (8 28 Dec 2021, 03:06
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Hi,

it doesn't have to be the median score necessarily.

The least distance is traveled when everyone decides to meet at a point which is between the 3rd and the 4th student. So any point between (87,0) and (142,0) would yield the same result. Just imagine the same question with 2, then 3, then 4 participants to see why.

Try it out with (100,0) and (142,0) as meeting point. Both yield 766. Again, to understand the reason, test it out with different number of participants
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Six children are standing along the x-axis at points (0,0), (30,0), (8 29 Mar 2023, 19:25
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It's much simpler. Pick a middle student such as 142 and sum the distances from the other students to 142.

0 -> 142
30 -> 112
87 -> 55
142 -> 0 (since doesn't have to move)
237 -> 95
504 -> 362

sum = 766
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Six children are standing along the x-axis at points (0,0), (30,0), (8 30 Mar 2023, 00:07
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nick1816
Six children are standing along the x-axis at points (0,0), (30,0), (87,0), (142,0), (237,0), (504,0). The children decide to meet at some point along the x-axis. What is the minimum total distance the children must walk in order to meet?

A. 504
B. 652
C. 766
D. 880
E. 1016
Show more

If there were say 3 children on a straight line at A(0, 0), D(142, 0) and F(504, 0), we know that for A and F to meet, they must together cover a distance of 504, no matter at which point they meet between 0 and 504.
Then it is best to make them meet at 142 so that D needs to cover no distance and total distance covered is minimized.

Based on this concept, when we have an odd number of children, we would like them to meet at the middle child's location. When we have an even number of children, we can make them meet at any point between the location of the middle two children.

A(0,0), B(30,0), C(87,0), D(142,0), E(237,0), F(504,0)

A and F need to cover 504 together and say they meet at some point between C and D.
B and E need to cover 207 ( = 237 - 30) together and say they meet at the same point between C and D.
C and D need to cover 55 ( = 142 - 87) together and they also meet at the same point between C and D.

So minimum distance covered by them = 504 + 207 + 55 = 766

Answer (C)
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Six children are standing along the x-axis at points (0,0), (30,0), (8 04 Jun 2024, 05:22
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A little mental arithmetic looking at the numbers will show that meeting at either end would be suboptimal and,obviously, past the ends even more so.

So they meet in between the ends, as our intuition on averages already suggests.

Set X as the distance from 0 that they meet.

For convenience, assume point X is between the lower three and the upper three. It doesn't affect the result, regardless.

So:

504-X
237-X
142-X
X-87
X-30
X

Adding these eliminates X, with the resulting sum being:

766

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Six children are standing along the x-axis at points (0,0), (30,0), (8 03 Jul 2025, 04:56
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