1)
How many people are there between the 4,000th person from the front end and the 1,000th person from the back end when there are 4,000 people on a line?
2)
There are 2,500 people on a line. The 100th person counted from the front is Adam. Adam is what number when he is counted from the back?
3)
How many people are there between the 1999th person counted from the front end and the 1999th person counted from the back end when there are 3499 people on a line?
4)
There are 2567 people on a line. How many people are there between the 497th person counted from the front end and the 1789th person counted from the back end?
{Please Help! Explain how you got the answers and show! Teacher was too lazy to explain}
Draw a line, and label the people and their distances.
#1 The 4000th person from the front is the last person. So, between that person and the 1000th from the last there are 999 people.
Now do some thinking and try the others. Why do I think that it is not (just) the teacher who was lazy? ...
1) In order to find out how many people are between the 4,000th person from the front end and the 1,000th person from the back end, we need to determine the total number of people between them.
When there are 4,000 people on a line, the person counting from the front end will start at 1 and end at 4,000. The person counting from the back end will start at 4,000 and end at 1.
So, the 4,000th person from the front end is the last person on the line, and the 1,000th person from the back end is the 3,001st person from the front end (since there are 3,000 people between the 3,000th and 4,000th person).
To find the number of people between them, we subtract the two positions. Therefore, the number of people between the 4,000th person from the front end and the 1,000th person from the back end is 4,000 - 3,001 = 999.
2) To determine Adam's position when counted from the back end, we need to consider that there are 2,500 people on the line.
If Adam is the 100th person counted from the front end, his position from the back end would be the 2,501st person.
To calculate this, we subtract Adam's position from the total number of people on the line: 2,500 - 100 = 2,400 (the number of people between Adam and the last person).
Then, we add 1 to this number to include Adam himself: 2,400 + 1 = 2,401.
Therefore, when Adam is counted from the back, he is the 2,401st person.
3) When there are 3,499 people on a line and we want to find out the number of people between the 1999th person counted from the front end and the 1999th person counted from the back end, we need to consider that the counting direction is reversed from both ends.
The 1999th person counted from the front end is the same as the 1,501st person counted from the back end (since there are 1,500 people between the 1,500th and 1,501st person).
To find the number of people between these two positions, we need to subtract the two positions: 1,501 - 1,999 = -498.
In this scenario, a negative number indicates that there are no people between the two positions.
Therefore, when there are 3,499 people on the line, there are no people between the 1999th person counted from the front end and the 1999th person counted from the back end.
4) In order to find out how many people are between the 497th person counted from the front end and the 1,789th person counted from the back end when there are 2,567 people on a line, we need to consider the counting direction from both ends.
The 497th person counted from the front end is the same as the 2,071st person counted from the back end (since there are 1,570 people between the 1,570th and 2,071st person).
To find the number of people between these two positions, we subtract the two positions: 2,071 - 497 = 1,574.
Therefore, when there are 2,567 people on the line, there are 1,574 people between the 497th person counted from the front end and the 1,789th person counted from the back end.