Question

In a potato race, a bucket is placed at the starting point, which is 5m from the first potato and the other potatoes are placed 3m apart in a straight line. There are 10 potatoes in the line. A competition starts from the bucket, pick up the nearest potato and run back with it, drop it in the next bucket, run back to pick up the next potato, run back to the bucket and drop it in and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

Answer 1

Hint: We are given, a competition is happening. We will first find what the first three terms means. We will find a distance run by her for picking the first 3 potatoes. Then, we get that they are forming arithmetic progression. Then we use ${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$ to find the total distance, as there are 10 potatoes so we will have n as 10.

Complete step-by-step solution:
We are given that, competitor has to collect the potatoes and then drop it back into the bucket. Means she first has to go from bucket to potato and then come back from potato to bucket.
So for the first potato, we know that, first potato is 5m away from the bucket, so she will run 5m to collect the potato and then run back 5m to drop the potato in the bucket.
Therefore, for the first potato she ran 10m.
For the second potato, we are given that the second potato is 3m away from the first potato. So she will run 5+3 = 8m to pick the potato and run back 5+3 = 8m to drop the potato. Therefore she runs 16m for the second potato.
For the third potato, the third potato is also 3m away from the second potato i.e. the third potato is 5+3+3 = 11m away from the bucket. Therefore she had to run 5+3+3 to pick the potato and then run back 5+3+3 to drop the potato. So in total she had to run 22m for the third potato.
So we can see that we are getting a series as 10, 16, 22 . . . . . . . . . . . . . . .
We can also see that the difference between terms is the same, 16-10 = 6 and 22-16 = 6.
So the series is moving by the same difference, and hence it is forming an arithmetic progression.
Now as the first term is 10 so we have a = 10.
And the difference between the series is 6 so we get d = 6.
Now to find the total run, we have to find the sum of these arithmetic sequence which is given as, ${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$.
We have values as a = 10, n = 10 and d = 6, so putting it in the formula we get,
${{S}_{10}}=\dfrac{10}{2}\left( 2\times 10+\left( 10-1 \right)6 \right)$.
Simplifying we get,
${{S}_{10}}=5\left( 20+\left( 9 \right)6 \right)$.
So we get,
${{S}_{10}}=5\times 74=370$
So she had to run 370m to pick and drop all the 10 potatoes.

Note: First of all remember that, when we do a product like a(b+c), we have to multiply first term by each term of the bracket i.e. $a\left( b+c \right)=ab+ac$.
Do not solve incorrectly as a(b+c) = ab + c.
Secondly we are given that, we have to pick and drop the potatoes, so we have to calculate the distance of picking along with the distance of dropping, else we would lead to a wrong answer.