How many people stand between

Learning outcome

You can count the people standing between two places, and you can read a swap back to the old place.

Concepts and reasons

When both places are counted from the same end, the people between them are the gap, not the difference. Gap = larger place minus smaller place, minus 1. A is 7th from the left and B is 18th from the left. Between them: 18 - 7 - 1 = 10. The 7 and the 18 are A and B, so they are not between themselves. The same gap works from the right. 9th and 4th from the right have 9 - 4 - 1 = 4 people between them. Places from opposite ends cannot be subtracted yet. First put both people on the same end. A swap does that. If A and B exchange places, the new left-place of A is the old left-place of B. A is 14th from the left. B is 8th from the right. After the swap, A is 20th from the left. So B used to be 20th from the left and 8th from the right. Total = 20 + 8 - 1 = 27.

Worked examples

1. 7th and 18th from the left. Between = 18 - 7 - 1 = 10.
2. 9th and 4th from the right. Between = 9 - 4 - 1 = 4.
3. A is 14th from the left and B is 8th from the right. After they swap, A is 20th from the left. B old left-place is 20. Total = 20 + 8 - 1 = 27. Between them on the left: 20 - 14 - 1 = 5.

Common traps

18 - 7 = 11 is the difference, not the people between. One of A or B is still inside that difference. Subtract 1 more. A left-place and a right-place are not a gap until both are left-places or both are right-places. After the swap, do not keep A at the old place. The new place is where B stood.

Practice questions

1. One person is 5th from the left and another is 12th from the left. How many stand between them?
2. A row has 30 people. How many stand between the 9th and the 22nd from the left?
3. A is 12th from the left. B is 7th from the right. After they swap, A is 15th from the left. How many people are in the row?

Answers and explanations

1. 6. 12 - 5 - 1 = 6.
2. 12. 22 - 9 - 1 = 12. The total is not needed once both places are from the left.
3. 21. The new place of A is the old left-place of B, so B was 15th from the left and 7th from the right. 15 + 7 - 1 = 21.

Analogy

Two house numbers on the same street: the houses between 5 and 12 are not 12 minus 5. Number 5 and number 12 are the two houses you started with. The doors between them are one fewer than the difference.

Quick reference

• Same end: between = larger place - smaller place - 1.
• Opposite ends: convert both to one end before you subtract.
• After a swap, the new left-place of A is the old left-place of B.
• Then total = that left-place + the right-place of B - 1.