Rank from both ends

Learning outcome

You can find how many people are in a line when one person has a rank from both ends, and how many people stand between two ranks counted from the same end.

Concepts and why

The same person is counted in both ranks, so the class size is rank from the top plus rank from the bottom, minus 1. When both places are counted from the same end, the people between them are the larger place minus the smaller place, minus 1. These rules are for a unique rank. They do not count a tie.

Worked examples

A student is 8th from the top and 15th from the bottom. Size = 8 + 15 - 1 = 22. Two students are 4th and 9th from the left. People between them = 9 - 4 - 1 = 4.

Common mistakes

Adding the two ranks and forgetting to subtract 1. That counts the same student twice. Subtracting the places and forgetting the last minus 1. That counts one of the two students as a person between them. Using this rule when the question says two students share a rank.

Practice questions

1. A student is 6th from the top and 10th from the bottom. How many students are in the class?
2. Two people are 3rd and 11th from the left. How many people stand between them?
3. A student is 1st from the top and 20th from the bottom. How many students are in the class?

Answers and explanations

1. 15. 6 + 10 - 1 = 15.
2. 7. 11 - 3 - 1 = 7.
3. 20. 1 + 20 - 1 = 20.

Analogy

A person who is both the first house from the east gate and the first house from the west gate is the only house. You do not add 1 and 1 to get 2. You subtract the double count.

Quick reference

Class size = top rank + bottom rank - 1. People between two places from the same end = larger place - smaller place - 1.