Let's analyze the problem step-by-step.
Problem Restatement
- There is a row of boys.
- Boy A is 10th from the left.
- Boy B is 9th from the right.
- They interchange their positions.
- After the interchange, A becomes x th from the left.
- We need to find the total number of boys in the row.
Step 1: Define variables
- Let the total number of boys be N.
- Initially:
- Position of A from the left = 10
- Position of B from the right = 9
- Since B is 9th from the right, B's position from the left = N−9+1=N−8N-9+1=N-8N−9+1=N−8.
Step 2: After interchange
- A moves to B's original position (which is N−8N-8N−8 from the left).
- So, after interchange, A becomes N−8N-8N−8th from the left.
Step 3: Use the given information
The problem states: "After interchange, A becomes th from the left."
This seems incomplete, but the key point is that after interchange, A's new
position from the left is the same as B's original position, which is
N−8N-8N−8.
Step 4: Relationship between positions
Since A was originally 10th from the left, and B was originally N−8N-8N−8th from the left, and after interchange, A takes B's position:
- A's new position from the left = N−8N-8N−8.
Step 5: Use the fact that B's position from the right is 9
- B's position from the right = 9
- So, N−B’s position from left+1=9N-\text{B's position from left}+1=9N−B’s position from left+1=9
- N−(N−8)+1=9N-(N-8)+1=9N−(N−8)+1=9
- 8+1=98+1=98+1=9 which is true, confirming the position of B.
Step 6: Find the total number of boys (N)
Since A's new position from the left is N−8N-8N−8, and A's original position from the left was 10, the problem implies:
- N−8=A’s new position from the leftN-8=\text{A's new position from the left}N−8=A’s new position from the left
- We know A's new position from the left is the position B had, so N−8N-8N−8 is the new position of A.
Step 7: Find the total number of boys
Since A's new position from the left is N−8N-8N−8, and A was originally 10th from the left, and B was originally 9th from the right:
- The total number of boys NNN satisfies:
A’s new position from left=N−8\text{A's new position from left}=N-8A’s new position from left=N−8
- But A's new position from the left is also the position B had, and B's position from the right is 9.
Step 8: Use the fact that after interchange, B becomes 10th from the left
(A's original position)
- After interchange, B takes A's original position, which is 10th from the left.
- So B's new position from the left = 10.
- B's original position from the right = 9.
Using this, total number of boys NNN can be calculated as:
B’s original position from left=N−9+1=N−8\text{B's original position from left}=N-9+1=N-8B’s original position from left=N−9+1=N−8
Given that B's new position from the left is 10, and B's original position from the left is N−8N-8N−8, the total number of boys NNN satisfies:
N−8=B’s original position from leftN-8=\text{B's original position from left}N−8=B’s original position from left
Final calculation:
Since B's original position from the left = N−8N-8N−8, and B's new position is 10th from the left, the total number of boys NNN is:
N−8=10N-8=10N−8=10
N=10+8=18N=10+8=18N=10+8=18
Answer:
There are 18 boys in the row. If you want, I can also provide a detailed explanation or a visual representation!
